††thanks: These two authors contributed equally.††thanks: These two authors
contributed equally.

# Influence of long range forces on the transition states and dynamics of NaCl
ion-pair dissociation in water

Dedi Wang  Biophysics Program and Institute for Physical Science and
Technology, University of Maryland, College Park 20742, USA.  Renjie Zhao
Chemical Physics Program and Institute for Physical Science and Technology,
University of Maryland, College Park 20742, USA.  John D.
Weeks111Corresponding author<EMAIL_ADDRESS>Department of Chemistry and
Biochemistry and Institute for Physical Science and Technology, University of
Maryland, College Park 20742, USA.  Pratyush Tiwary222Corresponding author.
<EMAIL_ADDRESS>Department of Chemistry and Biochemistry and Institute for
Physical Science and Technology, University of Maryland, College Park 20742,
USA.

###### Abstract

We study NaCl ion-pair dissociation in a dilute aqueous solution using
computer simulations both for the full system with long range Coulomb
interactions and for a well chosen reference system with short range
intermolecular interactions. Analyzing results using concepts from Local
Molecular Field (LMF) theory and the recently proposed AI-based analysis tool
“State predictive information bottleneck” (SPIB) we show that the system with
short range interactions can accurately reproduce the transition rate for the
dissociation process, the dynamics for moving between the underlying
metastable states, and the transition state ensemble. Contributions from long
range interactions can be largely neglected for these processes because long
range forces from the direct interionic Coulomb interactions are almost
completely canceled ($>90\%$) by those from solvent interactions over the
length scale where the transition takes place. Thus for this important
monovalent ion-pair system, short range forces alone are able to capture
detailed consequences of the collective solvent motion, allowing the use of
physically suggestive and computationally efficient short range models for the
disassociation event. We believe that the framework here should be applicable
to disentangling mechanisms for more complex processes such as multivalent ion
disassociation, where previous work has suggested that long range
contributions may be more important.

##  I Introduction

A wide variety of physical and biological processes such as protein
folding,Levy and Onuchic (2006); Dill (1990) self-assembly of dissolved
moleculesVenkateshwaran, Vembanur, and Garde (2014) and nucleation at
interfaces,Fang  _et al._ (2016) take place in aqueous environments. A concise
and accurate description of these systems requires an efficient treatment of
the interplay between the short and long range components of the
intermolecular interactions reflected in the local hydrogen bond network in
water and dielectric screening of distant solute charges.

Most computer simulations use periodic boundary conditions to address these
issues, where contributions from long range Coulomb interactions in periodic
images are determined by Ewald and related lattice sum methods.Allen and
Tildesley (2017); Belhadj, Alper, and Levy (1991); Essmann  _et al._ (1995)
However, this can introduce computation overhead due to their relatively poor
scaling with the system sizeSchulz  _et al._ (2009) and to parallel scaling
limitations from communication latency.Sun  _et al._ (2012); Kutzner  _et al._
(2014) Moreover, physical insight into the different roles of short and long
range interactions in equilibrium and dynamic properties is often obscured by
the indirect “black-box” nature of the lattice sum algorithms.

Indeed, the classic problem of ion-pair dissociation of NaCl in water has long
presented challenges to many simple pictures of solvent-mediated
processes.Smith and Dang (1994); Geissler, Dellago, and Chandler (1999);
Ballard and Dellago (2012); Mullen, Shea, and Peters (2014) Work by Geissler,
Dellago, and Chandler first demonstrated that the ion-pair distance $r_{ion}$
alone is not a sufficient reaction coordinate (RC) to describe the
dissociation dynamics, and reorganization of the surrounding solvent must be
taken into account.Geissler, Dellago, and Chandler (1999) Ballard and Dellago
showed that the dissociation event can be affected by solvent perturbations
extending out into the third solvation shell.Ballard and Dellago (2012)
However, a latter paper by Peters and coworkers Mullen, Shea, and Peters
(2014) argued that information only from water molecules in the first
solvation shell was needed to predict the committor, which is arguably a
perfect candidate for the RC and determines key features of the solvation
dynamics, such as the transition state ensemble.Bolhuis  _et al._ (2002)

In this paper we revisit this problem using some recent methods that we
believe can be naturally extended to more complicated solvent-mediated
processes such as protein conformational changes and crystal nucleation.
Rather than characterizing events in terms of solvent shell correlations, our
focus here is to determine the extent to which simplified and computationally
efficient reference models with well chosen short range intermolecular
interactions can give an accurate description of relevant static and dynamic
processes in the full long range system. As detailed in Sec. III, our results
show that short range interactions alone can reproduce most of the previously
determined dynamic properties of the NaCl ion-pair dissociation process
including the existence of solvent perturbations extending to the third
solvation shell and many features of the transition state ensemble.

Specifically, we employ a combination of Local Molecular Field (LMF)
theoryChen, Kaur, and Weeks (2004); Chen and Weeks (2006); Rodgers and Weeks
(2008a, b); Weeks (2002); Weeks, Chandler, and Andersen (1971) and our
recently proposed artificial intelligence (AI)-based analysis tool “State
predictive information bottleneck” (SPIB).Wang and Tiwary (2021) The LMF
treatment of aqueous solvated systems uses a minimal reference model for
disentangling the different effects of short and long range interactions on
equilibrium structural and thermodynamic properties, exploiting the near
cancellation of slowly varying long range Coulomb forces in uniform
environments. In previous work,Remsing, Rodgers, and Weeks (2011) the short
range Gaussian truncated (GT) water model, discussed in Sec. II below, was
shown to accurately reproduce atom-atom correlation functions of the bulk
water solvent and many properties of the hydrogen bond network, and it further
revealed the dominant role of short range interactions in the anomalous
behavior of the internal pressure and the density maximum.

Recently an LMF-based short solvent model was developed for predicting the
potential of mean force (PMF) between charged solutes in dilute aqueous
solutions.Gao, Remsing, and Weeks (2020) The short solvent model employs a
reference system where all Coulomb interactions involving solvent molecules
are Gaussian truncated and modified long range interactions are taken into
account only between solute charges. For the dilute NaCl solution of interest
here, it was previously found that these renormalized ion-ion corrections
could be neglected as well, with little deviation from the cation-anion PMF of
the full system.Gao, Remsing, and Weeks (2020) Thus a minimal short range
model where all Coulomb interactions are Gaussian truncated is sufficient to
describe the equilibrium PMF.

Here we investigate the extent to which the minimal model can also recover the
dissociation dynamics of NaCl by analyzing high-dimensional trajectories
generated from molecular dynamics (MD) simulations of full and short range
systems using insights from the SPIB method. SPIB approximates the RC as a
past-future information bottleneck, defined as the minimal amount of
information needed to be known about a system to accurately predict its future
state.

The SPIB approach has a number of useful features. First, it can automatically
partition the high-dimensional state space into a few metastable states with
minimal human intuition regarding the number or location of metastable states,
and also analyze the dynamics governing state-to-state movement. Second, it
can identify the correct transition state ensembles just given a limited
number of transitions. Third, inspired by Ref. Wang, Ribeiro, and Tiwary,
2019, it provides the use of a tunable hyper-parameter called the time-delay
$\Delta t$, which governs how far into the future predictions are being made.
By gradually increasing $\Delta t$ from 0, one can filter out all the fast
modes and control the level of coarse-graining of the system.

Through this combination of LMF and SPIB, we can carry out a systematic
kinetic analyses of NaCl ion-pair dissociation in water. We first study the
free energy barrier and the transition rate for the dissociation process. Then
we use SPIB to identify underlying metastable and transition states for the
dissociation process. The transitions between these metastable states and the
position of the transition state ensemble obtained by SPIB are then carefully
analyzed and compared to full system results.

##  II Methods

###  II.1 LMF theory and the Complete Gaussian Truncated solvent model

Building on ideas originally used for LJ potentials, Weeks, Chandler, and
Andersen (1971); Chandler, Weeks, and Andersen (1983) LMF theory asserts that
Coulomb intermolecular potentials in the full system of interest can be
usefully separated into strong short range and uniformly slowly-varying long
range components. To that end, we rewrite the potential $v(r)$ from a unit
point charge as

$v(r)=\frac{1}{r}=\frac{\operatorname{erfc}{(r/\bar{\sigma})}}{r}+\frac{\operatorname{erf}{(r/\bar{\sigma})}}{r}=v_{0}(r)+v_{1}(r),$
(1)

where erf and erfc are the error and complementary error functions and
$\bar{\sigma}$ is a smoothing or truncation length usefully chosen to be a
typical nearest neighbor distance in the full system.

The slowly varying component $v_{1}(r)$ in Eq.(1) is the potential of a unit
Gaussian change distribution whose width is proportional to $\bar{\sigma}$.
$v_{1}(r)$ approaches $1/r$ at large distances and remains slowly varying for
$r$ less than $\bar{\sigma}$ where strong intermolecular core forces exist. By
construction, $v_{0}(r)$ then accounts for the diverging short range Coulomb
force at distances less than $\bar{\sigma}$ and quickly vanishes for distances
larger than $\bar{\sigma}$.

Applying these ideas to charges in the SPC/E water model used for the solvent
in this paper, we define a Gaussian Truncated (GT) version by replacing the
point charge potentials by the short range $v_{0}(r)$, while retaining the
full oxygen LJ potential. Previous work has shown that GT water with any
choice of $\bar{\sigma}$ larger than the typical nearest-neighbor hydrogen
bond distances in water (roughly 3.5 Å) accurately reproduces the atom-atom
correlation functions of bulk SPC/E water along with many detailed features of
the local hydrogen bond network. Remsing, Rodgers, and Weeks (2011)

To clarify the role of short range forces for ion-pair systems solvated in
SPC/E water, we consider a simplified version of the short solvent model
discussed in the introduction where all Coulomb interactions in the system are
Gaussian truncated, including those between the ion pair charges. To determine
an appropriate $\bar{\sigma}$ for this minimal “Complete Gaussian truncated”
(CGT) solvent model, we note that the main influence of ions on the solvation
structure is in the first hydration shell. In particular, the Cl- anion can
also form hydrogen bonds with water molecules through the balance between the
strong short range electrostatic attraction and the repulsion involving its
somewhat larger LJ core. Taking this into account by making a minimal choice
of $\bar{\sigma}=5.0$ Å, we would expect both water-water and ion-water
hydrogen bonds to be well described by the CGT model.Remsing  _et al._ (2018)

Indeed, results using this CGT approximation in Ref. Gao, Remsing, and Weeks,
2020 showed that long range interactions in the NaCl system only negligibly
affected the equilibrium PMF. It is thus natural to ask if a similar situation
would persist for the corresponding dynamical process of ion-pair
dissociation. In this work, this is analyzed by comparing the dynamical
behavior of the CGT model, analyzed by SPIB, to that exhibited by the full
model, where Coulomb interactions are treated with conventional lattice sum
methods.

###  II.2 State Predictive Information Bottleneck (SPIB)

State predictive information bottleneck (SPIB) has been described in detail in
the original proof-of-principle publicationWang and Tiwary (2021) and here we
summarize its key features. A more complete introduction to SPIB and its
implementation details are discussed in Supplementary Material (SM). Similar
to the existing RAVE family of methods,Ribeiro  _et al._ (2018); Wang,
Ribeiro, and Tiwary (2019) SPIB aims to uncover a low dimensional manifold
(parameterized by a reaction coordinate or RC) on which the dynamics of the
system can be projected. We take $\bm{X}$ as a high-dimensional signal
characterizing a generic molecular system, and $\bm{y}$ as its corresponding
state label, which is drawn from a dictionary of indices for possible
metastable states. For the ion-pair dissociation problem considered here,
$\bm{X}$ could be expressed in terms of some order parameters, such as the
ion-pair distance, first and second shell coordination number and water
density, while $\bm{y}$ could comprise the contact state and dissociated
state. We describe these in detail in Sec. III.1. Let the values measured at
time $t$ be denoted by $\bm{X}_{t}$ and $\bm{y}_{t}$ respectively. Then we
assume that a good RC should carry the maximal predictive power for the future
state of the system. In other words, a desired RC should use minimal
information from the past signal $\bm{X}_{t}$ to predict its future state
label $\bm{y}_{t+\Delta t}$ accurately. Such a learning process can be easily
implemented through the deep variational information bottleneck
framework.Alemi  _et al._ (2016)

The workflow of SPIB is summarized as follows: First, we generate an initial
guess of the state labels. For ion-pair dissociation, this can be achieved by
simply dividing the input data space into a sufficiently fine grids along the
ion-pair distance as our initial state labels. We then input the trajectory
data $\bm{X}$ and the state labels $\bm{y}$ into SPIB, and try to find the
optimum RC which captures the most important features of the past
configuration $\bm{X}_{t}$ to predict the future state $\bm{y}_{t+\Delta t}$.
After this learning process, we can refine the state labels based on the
learned RC, and the new refined state labels are then fed back into SPIB. The
processes will be repeated until the converged RC and state labels are
generated for further analyses.

###  II.3 Simulation Methods

In our simulations, we consider a system composed of 727 SPC/E water molecules
and the NaCl ion-pair. Following Ref. Gao, Remsing, and Weeks, 2020, the ions
are modeled as point charges embedded in LJ cores with parameters
$\epsilon_{Na}=\epsilon_{Cl}=0.1$ kcal/mol, $\sigma_{Na}=2.586$ Å and
$\sigma_{Cl}=4.404$ Å, while the O- anion of the SPC/E water has LJ parameters
$\epsilon_{O}=0.15535$ kcal/mol and $\sigma_{O}=3.166$ Å. Lorentz–Berthelot
mixing rules Lorentz (1881) are used to determine the cross-interaction
parameters. Both the CGT model and the full model are simulated using the
LAMMPS package. Plimpton (1995)

The simulations are performed with a time step of 1 fs in the isothermal
isobaric (NPT) ensemble at $T=300$ K and $P=1$ atm, realized by a Nose-Hoover
thermostatNosé (1984) and barostatNosé and Klein (1983) with damping constants
of $0.1$ and $1.0$ ps, respectively. For each system, a 100 ns long
equilibrium trajectory is generated for further kinetic analyses. A total of
74 dissociation events through multiple transition pathways are observed in
the full system, while a total of 55 dissociation events are observed in the
CGT system. Further discussion of the two models is given in the SM.

##  III Results

###  III.1 Solvent Coordinates

Previous work has established that the collective motion of the solvent plays
an important role in the NaCl ion-pair dissociation.Geissler, Dellago, and
Chandler (1999); Ballard and Dellago (2012); Mullen, Shea, and Peters (2014)
Thus, by referring to Ref. Mullen, Shea, and Peters, 2014, in addition to the
distance between ions ($r_{ion}=|\bm{r}_{cation}-\bm{r}_{anion}|$), we
selected 12 other order parameters (OPs) to account for details of the solvent
arrangements. The definition of all the 13 OPs considered is provided in the
SM. All of these OPs depend only on the instantaneous local configuration of
solvent molecules, such as the coordination number up to the second solvation
shell. Among them, the number of bridging waters $N_{B}$ and the interionic
water density $\rho_{i3}$ are used for projection and visualization in the
main text to remain consistent with Ref. Mullen, Shea, and Peters, 2014. After
rescaling all the values into a range of $[0,1]$, the simulation trajectory
data in terms of these 13 OPs was used as the input $\bm{X}$ of SPIB to
analyze the kinetics of the NaCl ion-pair system.

###  III.2 Fundamental Kinetic Analysis

The cation–anion PMF calculated from the long unbiased simulation is shown in
Fig. 1(a). Though at least 3 basins can be seen along the distance between
ions ($r_{ion}$) based on the PMF, most workers have focused on the largest
barrier to dissociation.Smith and Dang (1994); Geissler, Dellago, and Chandler
(1999); Ballard and Dellago (2012); Mullen, Shea, and Peters (2014) Thus, the
barrier peak at about $r_{ion}=3.7$ Å and $3.6~{}k_{B}T$ is typically used to
distinguish the contact ion-pair from the solvent-separated ion-pair. The PMFs
obtained here are consistent with previous results from umbrella samplingGao,
Remsing, and Weeks (2020) with no significant differences in the PMFs,
especially in the first barrier to dissociation in the full system and CGT
system.

A further kinetic analysis of the transition rate also confirms this finding.
Based on the basin positions shown in Fig. 1(a), we define the contact state
as $r_{ion}<2.9$ Å and the solvent-separated state as $r_{ion}>5.0$ Å. Fig.
1(b) shows the distribution functions and their best-fit Poisson curves of the
mean first-passage time (MFPT) from the contact state to the dissociated state
for different systems. Maximum likelihood estimation (MLE) is used to estimate
the mean first-passage time and gives $\tau=20\pm 2\ ps$ for the full system
and $\tau=17\pm 2\ ps$ for the CGT system. Then we test the goodness-of-fit
based on the Kolmogorov–Smirnov (KS) statistic,Salvalaglio, Tiwary, and
Parrinello (2014) which gives $p\text{-value}=0.57$ and $0.62$, respectively.
This test result suggests that the mean first-passage time does follow a
Poisson distribution and that our state definition seems reasonable. The
transition time for the full system also agrees with the previous results from
transition path sampling and reactive flux calculations.Geissler, Dellago, and
Chandler (1999)

From this kinetic analysis, it seems plausible that the CGT system with all
Coulomb interactions truncated can reproduce basic features of the
dissociation kinetics of the NaCl ion-pair in water. Moreover, by leveraging
the power of SPIB, we can investigate the underlying dynamics in more detail.

Figure 1: (a) The potential of mean force (PMF) of the aqueous NaCl system as
a function of ion-pair distance $r_{ion}$. The PMF is calculated as
$-k_{B}T\log{g(r_{ion})}$, where $g(r)$ is the radial distribution function.
The local maximum of PMF at $r_{ion}=3.7$ Å separates the contact state
($r_{ion}<2.9$ Å) and dissociated state ($r_{ion}>5.0$ Å). (b) The mean first-
passage time (MFPT) from the contact state ($r_{ion}<2.9$ Å) to the
dissociated state ($r_{ion}>5.0$ Å ) for different systems. The dashed lines
denote the histogram data, while solid lines show corresponding best fits from
maximum likelihood estimation (MLE).

###  III.3 Metastable States Analysis

We now apply SPIB to the aqueous NaCl system where we do not assume any prior
knowledge about the system such as the number and location of metastable
states. In addition to $r_{ion}$, we introduce one solvent coordinate, the
number of bridging waters $N_{B}$, to better visualize the system as shown in
Fig. 2(a). In this case, we arbitrarily divide the input data space into a
fine grid used as our initial state labels (Fig. 2(b)).

Figure 2: The free energy surface (a) and the time delay dependent discrete-
state representation (b-e) for the full system projected onto coordinates
$r_{ion}$ and $N_{B}$. The initial state labels are shown in (b), while the
converged the state labels for different time delays are presented in (c-e).
The state labels are learned using the time delay $\Delta t=0.4ps$ (c),
$\Delta t=2.0ps$ (d), and $\Delta t=8.0ps$ (e) respectively. The color (or
state label) in each grid corresponds only to the state label with highest
fraction of samples for the respective grid point.

Fig. 2 reveals the underlying complexity in even such an apparently simple
system. In Fig. 2(c), we show how SPIB with a small time delay can
automatically identify the metastable states in this system (numbered state 1,
state 5, state 6 and state 9 by SPIB). The meaning of these metastable states
is summarized in Fig. 3(e). In addition to the contact state (state 1), the
solvent-separated state (state 6) and the completely dissociated state (state
9) shown on the profile of the potential of mean force (Fig. 1(a)), we can
identify another metastable state (numbered state 5). The existence of this
metastable state was discussed in Ref. Mullen, Shea, and Peters, 2014 as a
possible intermediate state that could play an important role in explaining
the recrossing behaviour of the ion-pair dissociation.

In fact, two additional intermediate states can be identified by SPIB using an
even smaller time delay ($\Delta t=0.1~{}ps$, see SM), but they have much
shorter lifetimes, so we will focus on the intermediate state 5 shown here
which has a much longer lifetime (about $1~{}ps$). We will still refer to this
newly identified state as an intermediate state in accordance with Ref.
Mullen, Shea, and Peters, 2014, though its state population and escape time
are in fact comparable to those of the solvent-separated state (see SM). This
intermediate state has very large overlap with the solvent-separated state in
the projection of $r_{ion}$, so it cannot be identified solely from the 1D PMF
in Fig. 1(a). But a clear correspondence can be observed between the four
metastable states uncovered by SPIB shown in Fig. 2(c) and the free energy
minima in the $r_{ion}$-$N_{B}$ space shown in Fig. 2(a). A more coarse-
grained representation of the ion-pair system can be obtained by controlling
the time delay $\Delta t$ as shown in Fig. 2(c-e). Here only the results for
the full system are presented, but the same time-dependent results are
obtained for the CGT system (see SM).

Figure 3: The multi-states dynamics learned by SPIB with the time delay
$\Delta t=0.4ps$ for the full system (a,c) and GT system (b,d). The top row
(a-b) are the state labels learned in the 2D RC space. The middle row (c-d)
shows the free energy surface (-$k_{B}T\log P(RC_{0},RC_{1})$) in the 2D RC
space. The bottom row (e) is the network representation of the transition
probability matrix for the four-state kinetics. The final converged state 1
corresponds to the contact state, state 5 corresponds to the intermediate
state, state 6 corresponds to the solvent-separated state, and state 9
corresponds to the completely dissociated state. The bridging water is
oriented with a hydrogen toward Cl- and the oxygen toward Na+ to compensate
the attraction force between the ion-pair in the intermediate state and the
solvent-separated state. The thickness of the arrows represents the transition
probability.

###  III.4 Interpreting the SPIB derived RC and constructing a Markov model

Since SPIB can filter out all the fast processes and retain only the relevant
slow dynamics of interest, here we can further compare the slow dynamics
learned by SPIB for the full interaction system and the CGT system (Fig. 3).
Fig. 3(a-d) indicate that similar 2D RCs can be learned by SPIB for these two
systems with the time delay $\Delta t=0.4ps$. Such a 2D RC can provide
detailed kinetic information about the metastable state dynamics. For
instance, the intermediate state (state 5), the solvent-separated state (state
6) and the dissociated state (state 9) are close to each other and separated
by very low barriers as shown in Fig. 3(c-d), while much higher barriers can
be observed between the contact state (state 1) and them. This suggests the
different interconversion timescales between these four metastable states.
Moreover, the 2D RC also clearly shows that there is no direct transition
between the intermediate state (state 5) and the dissociated state (state 9),
while a direct transition between the contact state (state 1) and the
dissociated state (state 9) exists.

To better interpret our RC, we adopt a method to identify the most important
input order parameters by sequential randomization of input parameters. Kemp,
Zaradic, and Hansen (2007); Jung, Covino, and Hummer (2019) We have provided
details of the procedure in SM. The top two important order parameters are the
ion-pair distance $r_{ion}$ and the number of bridging water molecules
$N_{B}$, which are used to visualize the different metastable states learned
by SPIB in Fig. 2. We also find that the learned RCs shared the same relevant
components for both the full and CGT systems. Thus, given the similar RCs
learned by SPIB, we are confident that removing all the long range
interactions will not significantly affect the underlying dynamics. The
implied timescale analysis further confirms that given the learned state
representations, only small differences are observed in the relaxation
timescales between the two systems (see SM).

Based on the learned state representation, a four state Markov model can be
built and the major kinetic pathways to dissociation are depicted in Fig.
3(e). Initially, bulk waters insert into the first solvation shell of Na+ and
Cl- and increase the total number of water molecules to 10 or 11 in
preparation for the transition (see SM). At the same time, one or two water
molecules orient to compensate the direct attractive force between the ion-
pair. The fluctuation along $r_{ion}$ allows a transition to the solvent-
separated state directly or via an intermediate state. The solvent-separated
state is also unstable and quickly transitions to the completely dissociated
state.

We found all the transitions are accompanied by the insertion of bulk water
into the first solvation shell, which leads to a total increase of 3 or 4
water molecules during the whole dissociation process. This finding agrees
with the previous report for this system.Ballard and Dellago (2012) As
mentioned earlier, even in the direct transition from the contact state to the
solvent-separated state or the completely dissociated state, there exist some
transient intermediate states. (See SM)

###  III.5 Transition State Ensemble Analysis

Building on the analysis of the metastable states for both the full and CGT
systems, we can use SPIB to further compare their transition state ensembles
(TSEs) for the ion-pair dissociation process. However, the existence of the
long-lived intermediate state identified in the last subsection makes it
unreasonable to study only the direct transitions between the contact state
and the solvent-separated state while ignoring all the intermediate states
between them. Thus, we continue to take a four-state representation with a
relatively small time delay ($\Delta t=0.4~{}ps$) as shown in Fig.2(c) to
describe the dissociation process and define the transition state ensemble as
the set of configurations whose transition probability to the contact state is
$0.5$.

Figure 4: Free energy projected onto coordinates $r_{ion}$ and $\rho_{i3}$
(top row), $r_{ion}$ and $N_{B}$ (bottom row) for different systems.
Transition state ensembles identified by SPIB with the time delay $\Delta
t=0.4ps$ are projected onto each surface as black points.

Fig. 4 shows the projection of the transition state ensembles identified by
SPIB onto the coordinates $r_{ion}$, $\rho_{i3}$ and $N_{B}$. The identified
transition states in both the full and CGT systems are similar to the results
of Ref. Mullen, Shea, and Peters, 2014. First, the higher the number of well-
oriented interionic waters $N_{B}$ that cancel the the direct interionic
force, the smaller is the $r_{ion}$ required to be in the transition states.
Second, all the transition states share an almost constant interionic water
density $\rho_{i3}$, as shown in the top panel in Fig. 4. To better interpret
the TSE obtained by SPIB, we extract all the samples near the transition
states ($0.01$ Å${}^{-3}<\rho_{i3}<0.018$ Å-3) for the input relevance
analysis (see SM). The top three relevant order parameters are the number of
bridging water molecules $N_{B}$, the interionic water density $\rho_{i4}$,
which contains information similar to $\rho_{i3}$, and the ion-pair distance
$r_{ion}$, consistent with the findings of Ref. Mullen, Shea, and Peters,
2014. Since the top 3 OPs in both systems are the same, this suggests that
similar transition state ensembles would be identified by SPIB. We provide
further details of the TSE analysis in SM.

###  III.6 Explaining why long range forces have little effect on NaCl
dynamics

Thus far, we have shown that the short range interactions alone can reproduce
many dynamical properties of NaCl ion-pair dissociation, including the
transition rate, the metastable state kinetics, and the transition state
ensemble. We now provide a mechanistic explanation for this peculiar finding.

To make a more detailed analysis, we measure their influence through the
average solvent force on the solute.Geissler, Dellago, and Chandler (1999);
Rey and Guardia (1992) We constrain the ion-pair distance $r_{ion}$ ranging
from $3.25$ to $4.45$ Å with a separation increment of $0.1$ Å to obtain 13
constrained ensembles for both the full and the CGT systems. Each constrained
trajectory is 1 ns long, and configurations are saved every 10 fs. We then use
the trained SPIB to identify the transition state ensemble (TSE) from the
saved configurations.

As shown in Fig. 5, we first confirm the previous finding of Ref. Geissler,
Dellago, and Chandler, 1999 that the opposing long range components of the
solvent force in the TSE along the interionic axis (blue squares) are
significantly different from the full system equilibrium results (blue dashed
line), and their ensemble average can cancel most effects of the direct
interionic interaction (blue solid line).

Figure 5: Mean solvent force on the ion-pair along the interionic axis as a
function of ionic separation $r_{ion}$ in different systems. The equilibrium
average (dashed line) and transition state ensemble average
(squares/triangles) are shown alongside the negative of the direct interionic
force (solid line). The standard deviation of the solvent force is reported to
show the actual fluctuation of the solvent force.

Moreover, Fig. 5 shows that in the complete absence of long range Coulomb
interactions, the direct interionic force in the CGT system is shifted by
$\sim 8.5~{}k_{B}T/\AA$ relative to the full system, while the solvent force
(in both equilibrium and the TSE) is shifted by a similar amount in the
opposite direction, such that the effective net force that controls the
relative motion between the ion pair is approximately unchanged when we adopt
the simplified CGT model. This is consistent with expectations from LMF theory
for the influence of the weak and uniformly slowly varying long ranged
components from the monovalent ion pair. As a result, the balance between the
solvent force and the direct interionic force in the TSE still holds for the
CGT system.

##  IV Conclusion

In this work we have explored the question of whether long range forces matter
for the transition state and dynamics of NaCl ion-pair dissociation in water.
For this purpose, here we have employed Local Molecular Field (LMF) based
framework to partition interactions into short range and long range
components. Our results illustrate that the short range interactions dominate
the dissociation process, and they alone are enough to reproduce the
transition rate for the dissociation process, the dynamics of the underlying
metastable states and the transition state ensemble.

Overall, in this work the scheme of separating short and long range
interaction enabled us to better understand the underlying mechanism of
dissociation dynamics in NaCl. While NaCl is an important system routinely
studied by many workers, as we show here it is still quite simple in the sense
of the minor role of the long ranged interactions. We do however think that
the combination of LMF and SPIB should be easily applicable in future work to
much more complicated systems. We are particularly interested in investigating
systems with multivalent ions, such as CaCl2, where the LMF corrections due to
stronger Coulomb tails may have more prominent influences. The LMF-based model
reduces the number of sites interacting through long range interactions to
that of only charged solute sites, thereby scaling almost linearly with the
size of the simulation cell. This feature makes the LMF-based model very
compatible with the advanced sampling methods which handle short range
interactions efficiently to further speed up the simulation. More
interestingly, the combination of the LMF-based models with the AI-based
advanced sampling methods, such as RAVE or SPIB,Ribeiro  _et al._ (2018);
Wang, Ribeiro, and Tiwary (2019); Wang and Tiwary (2021) may further
facilitate the determination of optimal short range reaction coordinates (RCs)
and generate some new insights into the specific physical problems of
interest.

Supplementary material
See supplementary material for details about the molecular dynamics
simulation, the implementation of SPIB, the definition of input order
parameters, and further analyses of kinetic data.

Acknowledgements
This research was entirely supported by the U.S. Department of Energy, Office
of Science, Basic Energy Sciences, CPIMS Program, under Award DE-SC0021009.
The authors thank Eric Beyerle for proof-reading the manuscript. This work
used the Extreme Science and Engineering Discovery Environment (XSEDE) Bridges
through allocation Grant No. CHE180027P, which is supported by the National
Science Foundation Grant No. ACI-1548562. We also thank MARCC’s Bluecrab HPC
clusters for computing resources.

Data availability statement
The data that support the findings of this study are available from the
corresponding author upon reasonable request.

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Let $\mathcal{F}=\{F_{\alpha}: \alpha\in \mathcal{A}\}$ be a family of infinite graphs, together with $\Lambda$. The Factorization Problem $FP(\mathcal{F}, \Lambda)$
asks whether $\mathcal{F}$ can be realized as a factorization of $\Lambda$, namely, whether
there is a factorization $\mathcal{G}=\{\Gamma_{\alpha}: \alpha\in \mathcal{A}\}$ of
$\Lambda$ such that each $\Gamma_{\alpha}$ is a copy of $F_{\alpha}$.

We study this problem when $\Lambda$ is either the Rado graph $R$
or the complete graph $K_\aleph$ of infinite order $\aleph$.
When $\mathcal{F}$ is a countable family,
we show that $FP(\mathcal{F}, R)$ is solvable if and only if
each graph in $\mathcal{F}$ has no finite dominating set.

We also prove that $FP(\mathcal{F}, K_\aleph)$ admits a solution
whenever the cardinality $\mathcal{F}$ coincide with the order and the domination numbers of its graphs.

For countable complete graphs, we show some non existence results when
the domination numbers of the graphs in $\mathcal{F}$ are finite. More precisely,
we show that there is no factorization of $K_\N$ into copies of a $k$-star (that is,
the vertex disjoint union of $k$ countable stars)  when $k=1,2$, whereas it exists when
$k\geq 4$, leaving the problem open for $k=3$.

Finally, we determine sufficient conditions for
the graphs of a decomposition to be arranged into
resolution classes.

Factorization Problem, Resolution Problem, Rado Graph, Infinite Graphs.
05C63, 05C70

§ INTRODUCTION

We assume that the reader is familiar
with the basic concepts in (infinite) graph theory, and refer to
[10] for further details.

In this paper all graphs will be simple, namely, without multiple edges or loops.
As usual, we denote by $V(\Lambda)$ and $E(\Lambda)$ the vertex set and the edge set of a simple graph $\Lambda$, respectively. We say that $\Lambda$ is finite (resp. infinite)
if its vertex set is so, and refer to the cardinality of $V(\Lambda)$ and $E(\Lambda)$
as the order and the size of $\Lambda$, respectively.
Note that in the finite case $|E(\Lambda)|\leq \binom{|V(\Lambda)|}{2}$, whereas if $\Lambda$ is infinite, then its order, which is a cardinal number, is greater than or equal to its size.
We use the notation $K_v$ for any complete graph of order $v$, and denote by $K_V$ the complete graph whose vertex set is $V$.

Given a subgraph $\Gamma$ of a simple graph $\Lambda$, we denote by
$\Lambda\setminus\Gamma$ the graph obtained from $\Lambda$ by deleting
the edges of $\Gamma$.
If $\Gamma$ contains all possible edges of $\Lambda$ joining any two of its vertices,
then $\Gamma$ is called an induced subgraph of $\Lambda$
(in other words, an induced subgraph is obtained by vertex deletions only).
Instead, if $V(\Gamma) = V(\Lambda)$, then $\Gamma$ is
called a spanning subgraph or a factor of $\Lambda$
(hence, a factor is obtained by edge deletions only).
If $\Gamma$ is also $h$-regular, then we speak of an $h$-factor.
We recall that a set $D$ of vertices of $\Lambda$ is dominating if all other vertices of $\Lambda$ are adjacent to some vertex of $D$. The minimum size of a dominating set of
$\Lambda$ is called the domination number of $\Lambda$. Finally, we say that
$\Lambda$ is locally finite if its vertex degrees are all finite.

A decomposition of $\Lambda$ is a set $\mathcal{G}=\{\Gamma_1,\ldots, \Gamma_n\}$ of subgraphs
of $\Lambda$ whose edge-sets partition $E(\Lambda)$.
If the graphs $\Gamma_i$ are all isomorphic to a given subgraph $\Gamma$ of $\Lambda$, then we speak of a
$\Gamma$-decomposition of $\Lambda$. When $\Gamma$ and $\Lambda$ are both complete graphs, we obtain $2$-designs. More precisely, a $K_k$-decomposition of $K_v$ is equivalent to a $2$-$(v,k,1)$ design.

Classically, the graphs $\Gamma_i$ and $\Lambda$ are taken to be finite,
and the same usually holds for the parameters $v$ and $k$ of a $2$-designs.
However, there has been considerable interest in designs on a infinite set of $v$ points,
mainly when $k=3$. In this case,  we obtain infinite Steiner triple systems whose first
explicit constructions were given in [13, 14].
Further results concerning the existence of rigid, sparse, and perfect countably Steiner triple systems can be found in [6, 7, 12]. Results showing that any Steiner system can be extended are given in [1, 16]. The existence of large sets of
Steiner triple systems for every infinite $v$ (and more generally, of infinite Steiner systems)
can be found in [4]. Also,
infinite versions of topics in finite geometry, including infinite Steiner triple systems and infinite perfect codes are considered in [3].
A more comprehensive list of results on infinite designs can be found in [9].

When each graph of a decomposition $\mathcal{G}$ of $\Lambda$ is a factor (resp. $h$-factor),
we speak of a factorization (resp. $h$-factorization) of $\Lambda$. Also, when the factors of
$\mathcal{G}$ are all isomorphic to $\Gamma$, we speak of a
$\Gamma$-factorization of $\Lambda$.
A factorization of $K_v$ into factors whose components are copies of $K_k$ is equivalent to
a resolvable $2$-$(v,k,1)$ design.

In this paper, we consider the Factorization Problem for infinite graphs, which is here stated in its most general version

Let $\Lambda$ be a graph of order $\aleph$ and let $\mathcal{F}=\{F_{\alpha}: \alpha\in \mathcal{A}\}$ be a family of (non-empty) infinite graphs,
not necessarily distinct, each of which has order $\aleph$, with $\aleph \geq |\mathcal{A}|$.

The Factorization Problem $FP(\mathcal{F}, \Lambda)$ asks for a factorization
$\mathcal{G}=\{\Gamma_{\alpha}:\alpha\in \mathcal{A}\}$ of $\Lambda$ such that $\Gamma_{\alpha}$ is isomorphic to $F_{\alpha}$,
for every $\alpha\in\mathcal{A}$.
If $\Lambda$ is the complete graph of order $\aleph$, we simply write $FP(\mathcal{F})$. If in addition to this each $F_\alpha$ is isomorphic to a given graph $F$ and $|\mathcal{A}|=\aleph$, we write $FP(F)$.
[Since in this case the factorization problem can be seen as a generalization of the Oberwolfach problem, in [8] the problem $FP(F)$ was denoted by $OP(F)$.]

As far as we know, there are only four papers dealing with the Factorization Problem for infinite complete graphs, and two of them, concern classic designs.
In [15] it is shown
that there exists a resolvable 2-design whenever $v=|\mathbb{N}|$ and $k$ is finite; these designs have, in addition, a cyclic automorphism group $G$ acting sharply transitive on the vertex set; briefly they are $G$-regular.
In [9] it is shown that every infinite $2$-design with $k<v$ is necessarily resolvable, and when $k=v$, both resolvable and non-resolvable designs exist.
We point out that both these papers deal with $t$-designs with $t\geq 2$.

in [2] the authors construct a $G$-regular 1-factorization of a countable complete graph
for every finitely generated abelian infinite group $G$.

Finally, [8] proves the following.
Let $F$ be a graph whose order is the cardinal number $\aleph$.
$FP(F)$ has a $G$-regular solution whenever the following two conditions hold:

*  $F$ is locally finite,

* $G$ is an involution free group of order $\aleph$.

Note that this result generalizes the one obtained in [15] to any complete graph of infinite order $\aleph$, blocks of any size less than $\aleph$, and groups $G$ not necessarily cyclic.
Furthermore, Theorem <ref> can also be seen as a generalization of the result in [2] to complete graphs of any infinite order.

When dealing with infinite graphs, a central role is played by the Rado graph $R$
(see [17]),
named after Richard Rado who gave one of its first explicit constructions. Indeed,
$R$ is the unique countably infinite random graph, and it can be constructed as follows:
$V(R)=\mathbb{N}$ and a pair $\{i,j\}$ with $i<j$ is an edge of $R$ if and only if
the $i$-th bit of the binary representation of $j$ is one. $R$ shows many interesting  properties, such as the universal property: every finite or countable graph can be embedded as an induced subgraph of $R$.

When replacing the concept of induced subgraph with the dual one of factor, a weaker result holds. Indeed, in [5] it is pointed out that a countable graph $F$
can be embedded as a factor of $R$ if and only if the domination number of $F$ is infinite. In the same paper, it is further shown that $FP(\mathcal{F}, R)$ has a solution whenever $\mathcal{F}$ is infinite and each of its graphs is locally finite.
Note that a locally finite countable graph has infinite domination number,
but the converse is not true:
for example, the Rado graph is not locally finite and it has no finite dominating set
(indeed, for every $D=\{i_1, \ldots, i_t\}\subset\mathbb{N}$, there exists an integer $j\in\mathbb{N}$ whose binary representation has 0 in positions $i_1, \ldots, i_t$, which means  that $j$ is adjacent with no vertex of $D$).

In this paper, we extend this result to any countable family $\mathcal{F}$ of admissible graphs. More precisely, we prove the following.

Let $\mathcal{F}$ be a countable family of countable graphs. Then,
$FP(\mathcal{F},R)$ has a solution if and only if the domination number of each graph of $\mathcal{F}$ is infinite.

Furthermore, we prove the solvability of $FP(\mathcal{F})$ whenever the size of
$\mathcal{F}$ coincides with the order and the domination number of its graphs.

Let $\mathcal{F}$ be a family of graphs, each of which has order $\aleph$.
$FP(\mathcal{F})$ has a solution whenever the following two conditions hold:

* $|\mathcal{F}|=\aleph$,

* the domination number of each graph in $\mathcal{F}$ is $\aleph$.

When $\mathcal{F}$ contains only copies of a given graph $F$ satisfying condition <ref> of Theorem <ref> (i.e., $F$ is locally finite), then $\mathcal{F}$ satisfies both conditions 1 and 2 of Theorem <ref>.
Therefore, Theorem <ref> can be seen as a generalization of Theorem <ref>, even though it does not provide any information on the automorphisms of a solution to FP.

Note that if we just require that the domination number of each graph of $\mathcal{F}$ is $\aleph$, there may exist factorizations with fewer factors than $\aleph$; this means that the two conditions in Theorem <ref> are independent.
Indeed, the Rado graph $R$ has no finite dominating set

and Corollary <ref> shows that for every $n\geq 2$ there exists a factorization of $K_\mathbb{N}$ into $n$ copies of $R$.
We point out that Theorem <ref> constructs instead  factorizations of $K_\mathbb{N}$ into infinite copies of $R$.

The paper is organized as follows.
In Sections 2 and 3, we prove the main results of this paper, Theorems <ref> and <ref>.

In Section 4, we deal with $F$-factorizations of $K_{\mathbb{N}}$ when $F$ belongs to a special class of graphs with finite domination number (and hence not satisfying condition <ref> of Theorem <ref>): the countable $k$-stars
(briefly, $S_k$), that is, the vertex disjoint union of $k$ countable stars.
We prove that $FP(S_k)$ has a solution whenever $k>3$, and there is no solution for $k\in \{1,2\}$.
This shows that there are families $\mathcal{F}$ of graphs for which
$FP(\mathcal{F})$ is not solvable. We leave open the problem when $k=3$.
In the last section, inspired by [9], we provide a sufficient condition for a decomposition $\mathcal{F}$ of $K_{\aleph}$ to be resolvable (i.e., the graphs of
$\mathcal{F}$ can be partitioned into factors of $K_{\aleph}$).

§ FACTORIZING THE RADO GRAPH

In this section, we prove Theorem <ref>. Also, since the Rado graph $R$ is self-complementary, that is,
$K_\mathbb{N}\setminus R$ is isomorphic to $R$, we obtain as a corollary
the countable version of Theorem <ref>.

We start by recalling an important characterization of the Rado graph
(see, for example, [5]).

A countable graph is isomorphic to the Rado graph if and only if it satisfies the following property:

$\star$ for every disjoint finite sets of vertices $U$ and $W$, there exists a vertex $z$ adjacent to all the vertices of $U$ and non-adjacent to all the vertices of $V$.

Now we slightly generalize the construction of the Rado graph given in the introduction.

Given a set $I\subset\{0,\dots,q-1\}$,
with $1 \leq |I| < q$, we denote by $R^q_{I}$ the following graph:
$V(R^q_{I})=\mathbb{N}$, and $\{x,y\}$, with $x<y$, is an edge of $R^q_{I}$ whenever the $x$-th digit of $y$ in the base $q$ expansion of $y$ belongs to $I$.

Cleraly, when $q=2$ and $I=\{1\}$
we obtain our initial description of the Rado graph (i.e. $R=R^2_{\{1\}}$).

Every graph $R^q_{I}$ is isomorphic to the Rado graph.

By Theorem <ref>, it is enough to show
that property $\star$ holds for $R^q_{I}$.
We assume, without loss of generality, that
$0\in I$ while $1\not\in I$, and let $U$ and $V$ be two disjoint subsets of $\mathbb{N}$. Then there are infinitely many positive integers whose base $q$ expansion has $0$ in each position $u\in U$
and $1$ in each position $v\in V$. Denoting by $z$ one of these integers larger than $\max(U \cup V)$, we have that
$z$ is adjacent to all the vertices of $U$ but to none in $V$.

Note that $K_{\mathbb{N}}= \bigcup_{i=0}^{q-1} R^q_{\{i\}}$
and $R^q_{\{0,\ldots,q-2\}}=\bigcup_{i=0}^{q-2} R^q_{i}$.
Considering that the $R^q_{\{i\}}$s are pairwise edge-disjoint and isomorphic to the Rado graph, we obtain the following.

For every $n\in\mathbb{N}$, the graphs $R$ and $K_{\mathbb{N}}$ can be factorized
into $n$ and $n+1$ copies of $R$, respectively.

The following result is crucial to prove Theorem <ref>. It strengthens a result given in [5] and allows us to suitably embed in the Rado graph $R$
any countable graph with infinite domination number.

Let $F$ be a countable graph with no finite dominating set.
For every edge $e\in E(R)$, there exists an embedding $\sigma_e$ of $F$ in $R$ such that:

*  $\sigma_e(F)$ is a spanning subgraph of $R$ containing the edge $e$;

*  $R\setminus \sigma_e(F)$ is isomorphic to $R$.

By Proposition <ref>, the graphs $R_{\{0,1\}}^3$,
$R_{\{0\}}^3$ and $R_{\{1\}}^3$ are isomoprhic to $R$.
Therefore, we can take $R= R_{\{0,1\}}^3$.

Let $e$ be an edge of $R  = R_{\{0\}}^3 \cup R_{\{1\}}^3$.
We can assume without loss of generality that
$e$ lies in $R_{\{0\}}^3$.
In  <cit.>,
it is shown that there exists an embedding
$\sigma_e$ of $F$ into the Rado graph $R_{\{0\}}^3 \subset R$
satisfying condition $1$. It is then left to
prove that condition <ref> holds. By Theorem <ref>, this is equivalent to saying that
$R\setminus \sigma_e(F)$ satisfies $\star$.

Let $U$ and $V$ be two finite disjoint subsets of $\mathbb{N}$.
Clearly, there are infinitely many positive integers whose base $3$ expansion has $1$ in each position $u\in U$ and $2$ in each position $v\in V$.
Let $z$ be one of these integers larger than $\max(U \cup V)$.
Since $R\setminus \sigma_e(F)$ contains $R_{\{1\}}^3$ and it is edge-disjoint with $R_{\{2\}}^3$,
it follows that  $z$ is adjacent in $R\setminus \sigma_e(F)$ to all the vertices of $U$ and is non-adjacent to all the vertices of $V$.
This means that $\star$ holds for $R\setminus \sigma_e(F)$.

We are now ready to prove Theorem <ref>, whose statement is recalled here, for clarity.

Theorem <ref>.
Let $\mathcal{F}$ be a countable family of countable graphs. Then, $FP(\mathcal{F},R)$ has a solution if and only if the domination number of each graph of $\mathcal{F}$ is infinite.

Since the Rado graph has no finite dominating set, the same holds for its spanning subgraphs. Hence, each graph of $\mathcal{F}$ must have infinite domination number. Under this assumption, we are going to show that $FP(\mathcal{F},R)$ has a solution.

Let $E(R)=\{ e_1,\dots,e_n,\ldots\}$ and
$\mathcal{F}=\{F_1,\dots,F_n,\ldots\}$. By recursively applying Proposition <ref>, we obtain a sequence
of isomorphisms $\sigma_i: F_i\rightarrow \Gamma_i$ satisfying
for each $i\in\N$ the following properties:

* $\Gamma_i$ is a spanning subgraph of $R$;

* $R\setminus (\Gamma_1\cup\Gamma_2\cup \cdots\cup \Gamma_{i-1})$
is isomorphic to $R$ and contains $\Gamma_i$;

* $e_i$ lies in $\Gamma_1\cup\Gamma_2\cup \cdots\cup \Gamma_i$.

It follows that the $\Gamma_i$s are pairwise edge-disjoint factors of $R$ which partition $E(R)$. Therefore,
$\{\Gamma_i: i\in \mathbb{N}\}$ is a solution to $FP(\mathcal{F},R)$.

The proof of Theorem <ref> allows us to construct solutions to $FP(\mathcal{F},R)$ even when
the cardinality of $\mathcal{F}$ is finite,
provided that $\mathcal{F}$ contains a copy of the Rado graph. In other words, we have the following.

Let $\mathcal{F}$ be a finite family of countable graphs such that

* $\mathcal{F}$ contains at least one graph isomorphic to the Rado graph;

* the domination number of each graph in $\mathcal{F}$ is infinite.

Then, $FP(\mathcal{F},R)$ has a solution.

Recalling that $R$ is self complementary, the countable version of Theorem <ref> can be easily obtained as a corollary to Theorem <ref>.

Let $\mathcal{F}$ be a countable family of countable graphs.
$FP(\mathcal{F})$ has a solution whenever the domination number of each graph in $\mathcal{F}$ is infinite.

Recall that $R_{\{0\}}^2$ and
$R_{\{1\}}^2$ are copies of $R$ which together factorize $K_{\mathbb{N}}$.   Therefore, it is enough to
partition $\mathcal{F}$ into two countable families $\mathcal{F}_1$ and $\mathcal{F}_2$, and then apply
Theorem <ref> to get a solution
$\mathcal{G}_i$ to $FP(\mathcal{F}_i, R_{\{i\}}^2)$, for $i=0,1$.
Clearly, $\mathcal{G}_1\cup \mathcal{G}_2$ provides a solution to  $FP(\mathcal{F})$.

The natural generalization of property $\star$ to a generic cardinality $\aleph$ is the following one:

$\star_{\aleph}$ for every disjoint sets of vertices $U$ and $W$ whose cardinality is smaller than $\aleph$, there exists a vertex $z$ adjacent to all the vertices of $U$ and non-adjacent to all the vertices of $V$.

Then, using the transfinite induction (see Theorem <ref> below), one could also prove the following generalization of Proposition <ref>:

Any two graphs of order $\aleph$ that satisfy property $\star_{\aleph}$ are pairwise isomorphic.

Therefore, we can refer to any graph of order $\aleph$
and satisfying property $\star_{\aleph}$
as the  $\aleph$-Rado graph $R_{\aleph}$.
Its existence is guaranteed under the
Generalized Continuum Hypothesis (GCH) which states that if
$\aleph'\prec \aleph$ then $2^{\aleph'}\preceq \aleph$.
Under GCH, one can see that the set of all $q$-ary sequences of length $\prec\aleph$
has size $\aleph$:  indeed, for every  $\aleph'\prec \aleph$, the set of all
$q$-ary sequences of length $\aleph'$ has cardinality $2^{\aleph'}$, and by GCH
we have that $2^{\aleph'}\preceq \aleph$.

This means that the construction of the countable Rado graph
(Definition <ref>) based on representing every natural number with a finite $q$-ary sequence (its base $q$ expansion) can be generalized to any order.

By assuming that GCH holds, we can prove the following generalization of Theorem <ref>.

Let $\mathcal{F}$ be a family of graphs of order $\aleph$
and assume that $|\mathcal{F}| = \aleph$.
Then $FP(\mathcal{F},R_{\aleph})$ has a solution if and only if
the domination number of each graph in $\mathcal{F}$ is $\aleph$.

§ FACTORIZING INFINITE COMPLETE GRAPHS

In this section we prove Theorem <ref>.
We point out that if we assume the Generalized Continuum Hypothesis,
considering that $R_\aleph$ is self complementary by property $\star_\aleph$,
we can proceed as in Corollary <ref>
and obtain Theorem <ref> as a consequence of
Theorem <ref>.

Here we present a proof of Theorem <ref>
that does not require GCH, which we recall is independent of ZFC.
Therefore, a proof that does not require GCH is to be preferred.

We say that a graph or a set of vertices is $\aleph$-small (resp. $\aleph$-bounded) if their order or cardinality is smaller than $\aleph$ (resp. smaller than or equal to $\aleph$).
Given two graphs $F$ and $\Lambda$ of order $\aleph$, we denote by $\Sigma_{\aleph}(F, \Lambda)$ the set of all graph embeddings
between an induced $\aleph$-small subgraph of $F$ and a subgraph of $\Lambda$.
A partial order on $\Sigma_{\aleph}(F, \Lambda)$ can be easily defined as follows:
$\sigma: G \rightarrow\Gamma$ and
$\sigma':G'\rightarrow\Gamma'$ are embeddings of $\Sigma_{\aleph}(F, \Lambda)$, we say that
$\sigma\leq \sigma'$ whenever $\sigma'$ is an extension of $\sigma$, namely,
$G$ and $\Gamma$ are subgraphs of  $G'$ and $\Gamma'$, respectively,
and $\sigma'|_{G} = \sigma$ (where $\sigma'|_{G}$ is the restriction of $\sigma'$ to $G$).

Let $F$ be a graph of order $\aleph$ and with no $\aleph$-small dominating set.
Also, let $\Theta$ be an $\aleph$-small subgraph of $K_{\aleph}$,
and let $\sigma\in \Sigma_{\aleph}(F, K_{\aleph}\setminus\Theta)$.

* If $v \in V(F)$, then there is an embedding $\sigma': G' \rightarrow \Gamma'$ in
$\Sigma_{\aleph}(F, K_{\aleph}\setminus\Theta)$ such that

\[|V(G')|\leq |V(G)|+1, \;\;\sigma\leq \sigma'\;\;\text{and}\;\; v\in V(G');\]

* If $x \in V(K_{\aleph})$, then there is an embedding $\sigma'': G'' \rightarrow \Gamma''$ in
$\Sigma_{\aleph}(F, K_{\aleph}\setminus\Theta)$ such that

\[|V(G'')|\leq|V(G)|+1, \;\; \sigma\leq \sigma''\;\;\text{and}\;\; x\in V(\Gamma'').\]

Let $\sigma: G \rightarrow \Gamma$ be an embedding in
$\Sigma_{\aleph}(F, K_\mathbb{N}\setminus\Theta)$, and let $v\in V(F)$ and $x\in V(K_{\aleph})$.
Clearly, when $v\in V(G)$ or $x\in V(\Gamma)$,
we can take $\sigma'=\sigma$ or $\sigma''=\sigma$, respectively. Therefore, we can assume $v\not \in V(G)$ and $x\not\in V(\Gamma)$.

* Let $G'$ be the subgraph of $F$ induced by $v$ and $V(G)$.
Since $V(\Theta)$ is $\aleph$-small, we can  choose $a\in V(K_{\aleph})\setminus V(\Theta)$ and
let $\sigma': V(G)\ \cup \{v\} \rightarrow V(\Gamma)\ \cup\ \{a\}$ be the extension of
$\sigma$ such that $\sigma'(v)=a$.  Setting $\Gamma' = \sigma'(G')$, we have that
$\sigma'$ is the required embedding of $\Sigma_{\aleph}(F, K_{\aleph}\setminus\Theta)$.

* Since $F$ has no $\aleph$-small dominating set, $V(G)$ (which is an $\aleph$-small set)
cannot be a dominating set for $F$.
Hence, there is a vertex $a\in V(F)$ that is not adjacent to any of the vertices
of $G$. We denote by $G''$ (resp., $\Gamma''$) the graph obtained by adding
$a$ to $G$ (resp., $x$ to $\Gamma)$ as an isolated vertex. Clearly, $G''$ is an induced subgraph of $F$; also, $\Gamma''$ and $\Theta$ have no edge in common,
since $E(\Gamma'')= E(\Gamma)$.
Therefore, the extension $\sigma'': G'' \rightarrow \Gamma''$ of $\sigma$
such that $\sigma''(a)=x$ is the required embedding of
$\Sigma_{\aleph}(F, K_\aleph\setminus\Theta)$.

From now on, we will work within the Zermelo-Frankel axiomatic system with the Axiom of Choice in the form of the Well-Ordering Theorem. We recall the definition of a well-order.

A well-order $\prec$ on a set $X$ is a total order on $X$ with the property that every non-empty subset of $X$ has a least element.

The following theorem is equivalent to the Axiom of Choice.

Every set $X$ admits a well-order $\prec$.

Given an element $x\in X$, we define the section $X_{\prec x}$ associated to it:

$$X_{\prec x}=\{y\in X: y\prec x\}.$$

Every set $X$ admits a well-order $\prec$ such that the cardinality of any section is smaller than $|X|$.

Let us consider a well-order $\prec$ on $X$. Let $x$ be the smallest element such that $X_{\prec x}$ has the same cardinality as $X$. The set $Y=X_{\prec x}$ is such that all its sections with respect to the order $\prec$ have smaller cardinality. Since $Y$ instead has the same cardinality as $X$, the order $\prec$ on $Y$ induces an order $\prec'$ on $X$ with the required property.

We recall now that well-orderings allow proofs by induction.

[Transfinite induction]
Let $X$ be a set with a well-order $\prec$ and let $P_x$ denote a property for each
$x\in X$. Set $0=\min  X$ and assume that:

* $P_0$ is true, and

* for every $x\in X$, if $P_y$ holds for every $y\in X_{\prec x}$, then $P_x$ holds.

Then $P_x$ is true for every $x\in X$.

We are now ready to prove Theorem <ref>.
The idea behind the proof can be better understood by restricting our attention to the countable case, $\aleph = \mathbb{N}$.
To solve $FP(\{F_\alpha: \alpha\in \mathbb{N}\})$,
we first order the edges of $K_\mathbb{N}: \{e_0, e_1, \ldots, e_\gamma, \ldots\}$.
Then, we define
embeddings $\sigma_\alpha^\beta: G_\alpha^\beta\rightarrow \Gamma_\alpha^\beta$
where $G_\alpha^\beta$ is an induced subgraph of $F_\alpha$,
and $\Gamma_\alpha^\beta$ is a subgraph of $K_{\mathbb{N}}$.
These embeddings are obtained by recursively applying Lemma <ref>
which adds, at each step, a vertex to  $G_\alpha^\beta$ and  a vertex to  $\Gamma_\alpha^\beta$
and makes sure that the vertex $\beta$ belongs to both these graphs
(this procedure can be seen as a variation of Cantor's “back-and-forth” method).
We also make sure that, for every $\gamma$,
the graphs $\Gamma_0^\gamma, \Gamma_1^\gamma, \ldots, \Gamma_\gamma^\gamma$ are pairwise edge-disjoint
and contain between them the edge $e_\gamma$.
The solution to $FP(\{F_\alpha: \alpha\in \mathbb{N}\})$ will be represented by
$\mathcal{G}=\{\Gamma_\alpha: \alpha\in \mathbb{N}\}$ where $\Gamma_\alpha=\bigcup_{\beta} \Gamma_\alpha^\beta$.

Theorem <ref>.

Let $\mathcal{F}$ be a family of graphs, each of which has order $\aleph$.
$FP(\mathcal{F})$ has a solution whenever the following two conditions hold:

* $|\mathcal{F}|=\aleph$,

* the domination number of each graph in $\mathcal{F}$ is $\aleph$.

Let $\mathcal{F} = \{F_\alpha:\alpha\in\mathcal{A}\}$.
We consider a well-order $\prec$ on $\mathcal{A}$ satisfying Corollary <ref>.
Since by assumption $|V(F_{\alpha})|=|\mathcal{A}|=\aleph$, for every $\alpha\in \mathcal{A}$, we can take $V(F_{\alpha})=V(K_{\aleph})= \mathcal{A}$ and index the edges of $K_{\aleph}$ over $\mathcal{A}$: $E(K_{\aleph})=\{e_{\alpha}: \alpha\in  \mathcal{A}\}$.

To prove the assertion, we construct a chain of families
$(\mathcal{E}_{\gamma})_{\gamma\in \mathcal{A}}$, where

\[
\mathcal{E}_{\gamma}:=\{\sigma_\alpha^\beta: G_\alpha^{\beta} \rightarrow \Gamma_\alpha^\beta\mid  \sigma_\alpha^\beta \in \Sigma_{\aleph}(F_\alpha, K_{\aleph}),(\alpha,\beta) \in \mathcal{A}_{\preceq \gamma}\times\mathcal{A}_{\preceq \gamma}\},
\]

which satisfy the ascending property, that is,
$\mathcal{E}_{\gamma'}\subseteq \mathcal{E}_{\gamma}$ if $\gamma'\preceq \gamma$,
and the following three conditions:

($1_\gamma$) for every $(\alpha,\beta)\in\mathcal{A}_{\preceq \gamma}\times\mathcal{A}_{\preceq \gamma}$ and $\beta'\prec \beta$ we have that $\sigma_{\alpha}^{\beta'}\leq \sigma_{\alpha}^{\beta}$ and $\beta\in V(G^{\beta}_{\alpha})\ \cap \ V(\Gamma^{\beta}_{\alpha})$;

($2_\gamma$) for every $\beta\in\mathcal{A}_{\preceq \gamma}$, the graphs
$\Gamma_{\alpha}^\beta: \alpha\preceq \beta $ are pairwise edge-disjoint, and the edge $e_\beta$ belongs to their union;

($3_\gamma$) for every $\alpha,\beta\in\mathcal{A}_{\preceq \gamma}$, the graph
$\Gamma_{\alpha}^\beta$ is either finite or $|\mathcal{A}_{\preceq \gamma}|$-bounded.

The desired factorization of $K_\aleph$ is then
$\mathcal{G}=\{\Gamma_\alpha: \alpha\in \mathcal{A}\}$,
where $\Gamma_\alpha =\bigcup_{\beta\in\mathcal{A}} \Gamma^\beta_\alpha$ for every $\alpha\in\mathcal{A}$.
Indeed, properties ($1_\gamma$) guarantee that each $\Gamma_\alpha$ is a factor of $K_\aleph$ isomorphic to $F_\alpha$. Also, properties ($2_\gamma$) ensure that the $\Gamma_\alpha$s are pairwise edge-disjoint and between them
contain all the edges of $K_{\aleph}$.

We proceed by transfinite induction on $\gamma$.

BASE CASE. Let $0=\min  \mathcal{A}$,
choose an edge $e\in E(F_{0})$ and let $\sigma\in \Sigma_\aleph(F_{0}, K_\aleph)$
be the embedding that maps $e$ to $e_{0}$. By Lemma <ref>, there exists an embedding
$\sigma_0^0: G_0^0 \rightarrow \Gamma_0^0$ in
$\Sigma_\aleph(F_{0}, K_\aleph)$ such that $\Gamma_0^0$ is a finite graph and

\[ \sigma\leq \sigma_0^0\;\;\text{and}\;\; 0\in
V(G_0^0)\cap V(\Gamma_0^0).
\]

Clearly, $\mathcal{E}_0:=\{\sigma_{0}^0\}$ satisfies properties ($1_0$), ($2_0$) and ($3_0$).

TRANSFINITE INDUCTIVE STEP.
We assume that, for any $\gamma'\prec \gamma$, there is a family $\mathcal{E}_{\gamma'}$ satisfying properties ($1_{\gamma'}$), ($2_{\gamma'}$) and ($3_{\gamma'}$), and
prove that it can be extended to a family $\mathcal{E}_{\gamma}$ that satisfies properties ($1_{\gamma}$), ($2_{\gamma}$) and ($3_{\gamma}$).
Clearly it is enough to provide the maps $\sigma_\alpha^\beta$ where either $\alpha=\gamma$ or $\beta=\gamma$.

We start by constructing the maps $\sigma_{\alpha}^{\gamma}$ for every $\alpha\prec \gamma$.
We proceed by transfinite induction on $\alpha$.

* Base case. Set $\Theta_0:=\bigcup_{\alpha, \beta\prec \gamma} \Gamma_\alpha^\beta$ and note that, by property ($3_{\gamma'}$), $\Theta_0$ is $\aleph$-small. We also set $\sigma_{0}^{\prec\gamma}: \bigcup_{\beta\prec \gamma} G_0^{\beta} \rightarrow \bigcup_{\beta\prec \gamma}  \Gamma_0^{\beta}$ to be the map of
$\Sigma_{\aleph}(F_0, K_\aleph\setminus\Theta_0)$ whose restriction to $G_0^{\beta}$ is $\sigma_0^{\beta}$. We note that property ($3_{\gamma'}$) guarantees that the order of $\bigcup_{\beta\prec \gamma} G_0^{\beta}$ is either finite or $|\mathcal{A}_{\preceq \gamma}|$-bounded,
hence $\aleph$-small.

Therefore, we can apply Lemma <ref>
(with  $\sigma = \sigma_{0}^{\prec\gamma}$) to obtain
the map $\sigma_0^{\gamma}: G_0^{\gamma} \rightarrow \Gamma_0^{\gamma}$ in
$\Sigma_{\aleph}(F_0, K_\aleph\setminus\Theta_0)$ such that $|V(\Gamma_0^{\gamma})|\leq|V(\bigcup_{\beta\prec \gamma} \Gamma_0^{\beta})|+2$ and, for every $\gamma'\prec \gamma$,

\[ \sigma_0^{\gamma'}\leq \sigma_0^{\gamma}\;\;\text{and}\;\; \gamma\in V(G_0^{\gamma})\cap V(\Gamma_0^{\gamma}).\]

* Inductive step. Assume we have defined the maps
$\sigma_{\alpha'}^{\gamma}$ for every $\alpha'\prec \alpha$, and set

\[
\Theta_\alpha:=\bigcup_{\alpha'\prec \alpha} \Gamma_{\alpha'}^{\gamma}\;\cup\;
\bigcup_{\alpha \prec\alpha'\prec \gamma, \beta \prec \gamma} \Gamma_{\alpha'}^{\beta}.
\]

As before,  by Lemma <ref> there exists $\sigma_\alpha^{\gamma}: G_\alpha^{\gamma} \rightarrow \Gamma_\alpha^{\gamma}$ in
$\Sigma_{\aleph}(F_\alpha, K_\aleph\setminus\Theta_\alpha)$ such that $|V(\Gamma_\alpha^{\gamma})|\leq|V(\bigcup_{\beta\prec \gamma} \Gamma_\alpha^{\beta})|+2$ and, for every $\gamma'\prec \gamma$,

\[
\sigma_\alpha^{\gamma'}\leq \sigma_\alpha^{\gamma}\;\;\text{and}\;\;\gamma\in V(G_\alpha^{\gamma})\cap V(\Gamma_\alpha^{\gamma}).
\]

Finally, we define the maps $\sigma_{\gamma}^\beta$ when $\beta \preceq \gamma$. We set $\Theta:=\bigcup_{\alpha \prec \gamma} \Gamma_\alpha^{\gamma}$ and proceed by transfinite induction on $\beta$.

* Base case.
If $e_{\gamma}\in \Theta$, let $\sigma$ be the empty map of $\Sigma_{\aleph}(F_{\gamma}, K_\aleph\setminus\Theta)$. Otherwise, chose an edge $e\in E(F_{\gamma})$,
and let $\sigma\in \Sigma_{\aleph}(F_{\gamma}, K_\aleph\setminus\Theta)$
be the embedding that maps $e$ to $e_{\gamma}$.
By Lemma <ref>, there exists $\sigma_{\gamma}^0: G_{\gamma}^0 \rightarrow \Gamma_{\gamma}^0$ in
$\Sigma_{\aleph}(F_{\gamma}, K_\aleph\setminus\Theta)$ such that $\Gamma_{\gamma}^0$ is a finite graph and

\[ \sigma\leq \sigma_{\gamma}^0\;\;\text{and}\;\; 0\in V(G_{\gamma}^0)\cap V(\Gamma_{\gamma}^0).\]

* Inductive step.
Assume we have defined the maps $\sigma_{\gamma}^{\beta'}$ for any $\beta'\prec \beta$. Again by Lemma <ref>, there exists $\sigma_{\gamma}^{\beta}: G_{\gamma}^{\beta} \rightarrow \Gamma_{\gamma}^{\beta}$ in
$\Sigma_{\aleph}(F_{\gamma}, K_\aleph\setminus\Theta)$ such that $|V(\Gamma_\gamma^{\beta})|\leq|V(\bigcup_{\beta'\prec \beta} \Gamma_\gamma^{\beta'})|+2$ and, for any $\beta'\prec \beta$,

\[ \sigma_{\gamma}^{\beta'}\leq \sigma_{\gamma}^{\beta}\;\;\text{and}\;\;
\beta\in
V(G_{\gamma}^{\beta})\cap V(\Gamma_{\gamma}^{\beta}).
\]

It follows from the construction that the family
\[\mathcal{E}_{\gamma}:=\{\sigma_\alpha^\beta: G^\alpha_{\beta} \rightarrow \Gamma_\alpha^\beta\mid  \sigma_\alpha^\beta \in \Sigma_\aleph(F_\alpha, K_\aleph), \alpha,\beta\preceq \gamma\}\]

satisfies properties $(1_\gamma)$, $(2_\gamma)$ and $(3_\gamma)$.

§ THE FACTORIZATION PROBLEM FOR $K$-STARS

Theorem <ref> does not provide solutions to $FP(F)$ whenever the graph $F$ has a dominating set
of cardinality less than its order. In particular, if $F$ is countable with a finite dominating set, then the existence of a solution to $FP(F)$ is an open problem. In this section, we consider a special class of such graphs, the $k$-stars $S_k$. More precisely,

* the star $S_1$ is the graph with vertex-set $\mathbb{N}$ whose edges are of the form
$\{0, i\}$ for every $i\in \mathbb{N}\setminus \{0\}$;

* the $k$-star $S_{k}$ is the vertex-disjoint union of $k$ stars.

Note that $S_k$ contains exactly $k$ vertices of infinite degree, which we call centers
and form a finite dominating set of $S_k$.

In the following, we show that $FP(S_k)$ has no solution whenever $k\in \{1,2\}$, while it admits a solution for every $k>3$. Unfortunately, we leave open the problem for $3$-stars.

§.§ The case $k\in \{1,2\}$

$FP(S_1)$ has no solution.

Assume for a contradiction that there is a factorization $\mathcal{G}$ of $K_{\mathbb{N}}$ into $1$-stars. Choose any star $\Gamma\in \mathcal{G}$ and let $g$ denote its center. Considering that
all the edges of $K_{\mathbb{N}}$ incident with $g$ belong $\Gamma$, it follows that $g$ cannot be a vertex in any other star of $\mathcal{G}$, which therefore are not factors and this is a contradiction.

With essentially the same proof, one obtains the following.

Let $F$ be the vertex-disjoint union of $S_1$ with a finite set of isolated vertices.
Then $FP(F)$ has no solution.

To prove the non-existence of a solution to $FP(S_2)$ it will be useful the following lemma.

If $\mathcal{G}$ is a factorization of $K_{\mathbb{N}}$ into $k$-stars, then
there is at most one vertex of $K_{\mathbb{N}}$ that is never a center in any $k$-star of $\mathcal{G}$. It follows that $|\mathcal{G}|=|\mathbb{N}|$.

It is enough to notice that every pair $\{a,b\}$ of vertices of
$K_{\mathbb{N}}$ is the edge of some $2$-star $\Gamma$ of $\mathcal{G}$;
hence, either $a$ or $b$ is a center of $\Gamma$.

$FP(S_2)$ has no solution.

Assume for a contradiction that there is a factorization $\mathcal{G}$ of $K_{\mathbb{N}}$ into $2$-stars. For every $\Gamma\in\mathcal{G}$, letting $c$ be a center of $\Gamma$, we denote by
$\Gamma(c)$ the set of vertices adjacent with $c$ in $\Gamma$ (i.e., the neighborhood of $c$ in $\Gamma$).

Choose any $2$-star $\Gamma\in \mathcal{G}$ and let $a$ and $b$ denote its centers.
Also, let $\Gamma'$ be the $2$-star of $\mathcal{G}\setminus\{\Gamma\}$ containing the edge $\{a,b\}$.
Without loss of generality, we can assume that $a$ is a center of $\Gamma'$.
Finally, by Lemma <ref>, we can choose $x\in \Gamma'(a)\setminus\{b\}$
such that there exists a $2$-star  $\Gamma''\in \mathcal{G}$ having $x$ as a vertex.

Since $\Gamma$ is a factor of
$K_{\mathbb{N}}$, it follows that $x\in \Gamma(b)$. In other words,
$\Gamma\ \cup\ \Gamma'$ contains the edges $\{x,a\}$ and $\{x,b\}$. Therefore,
$a,b\not\in\Gamma''(x)$. Since $\Gamma''$ is a factor of $K_{\mathbb{N}}$
and $\{a,b\}$ is an edge of $\Gamma$, it follows
$a,b\in \Gamma''(y)$, where $y$ is the other center of $\Gamma''$.
In other words, $\{y,a\}$ and $\{y,b\}$ belong to $\Gamma''$, hence $y$
cannot lie in $\Gamma$, contradicting the fact that $\Gamma$ is a factor.

§.§ The case $k\geq 4$

In this section we prove the solvability of $FP(S_k)$ whenever $k\geq 4$.
For our constructions we need to introduce the following notation.

Let $\mathbb{D}$ be an integral domain and set $V=\mathbb{D}\times \{0,1,\ldots,h\}$, for $h\geq 0$.
For the sake of brevity, we will denote each pair $(a,i)\in V$ by $a_i$.
Given a graph $\Gamma$ with vertices in $V$, for every $a,b\in \mathbb{D}$ we denote by
$a\Gamma+b$ the graph obtained by replacing each vertex $x_i$ of $\Gamma$ with $(ax+b)_i$.
Also, we denote by $Orb_{\mathbb{D}}(\Gamma)=\{\Gamma+d:d\in \mathbb{D}\}$ the
$\mathbb{D}$-orbit of $\Gamma$, that is, the set of all translates of $\Gamma$ by the elements of

For every $k\geq 4$, there exists a $k$-star $\Gamma$ with vertex set $V=\mathbb{Z}\times\{0,1\}$ such that $Orb_{\mathbb{Z}}(\Gamma)$ is a factorization of $K_{V}$ into $k$-stars.

We first deal with the case $k=4$. Set $\Gamma= \bigcup_{i=1}^4 \Gamma_i$, where
each $\Gamma_i$ is the $1$-star with vertices in  $V=\mathbb{Z}\times\{0,1\}$
and center $x_i$ defined as follows (see Figure <ref>):

* $x_1= 0_0$ and $\Gamma_1(x_1)=\{i_0: i\geq 1\}$;

* $x_2= -1_1$ and $\Gamma_2(x_2)=\{i_1: i\geq 0\}\ \cup\ \{-1_0\}$;

* $x_3= -2_0$ and $\Gamma_2(x_3)=\{i_1: i\leq -3\}$;

* $x_4= -2_1$ and $\Gamma_4(x_4)=\{i_0: i\leq -3\}$.

We claim that  $\mathcal{G}:=Orb_{\mathbb{Z}}(\Gamma)$ is a factorization of $K_V$ into $4$-stars.
Denote by
$K_{U,W}$ the complete bipartite graph whose parts are $U=\mathbb{Z}\times\{0\}$ and $W=\mathbb{Z}\times \{1\}$, and consider the $1$-factor $I=\{\{i_0, i_1\}:i\in\mathbb{Z}\}$ of $K_{U,W}$.
Clearly, $K_V$ decomposed into $K_U$, $K_W  \cup I$ and $K_{U,W}\setminus I$.
One can check that

* $Orb_{\mathbb{Z}}(\Gamma_1)$ decomposes $K_U$,

* $Orb_{\mathbb{Z}}(\Gamma_2)$ decomposes $K_W  \cup I$, and

* $Orb_{\mathbb{Z}}(\Gamma_3\cup\Gamma_4)$ decomposes $K_{U,W}\setminus I$.

Hence, $\mathcal{G}$ is a decomposition of $K_V$. Considering that
the $\Gamma_i$s are pairwise vertex-disjoint and their vertex-sets partition $V$, we have that
$\Gamma$ and each of its translates (under the action of $\mathbb{Z}$)
are factors of $K_V$ isomorphic to a $4$-star. Therefore,
$\mathcal{G}$ is a factorization  of $K_V$ into $4$-stars.

To deal with the case $k\geq 5$, it is enough to replace the component $\Gamma_1$ of $\Gamma$ with a
$(k-3)$-star $\Gamma'_1$ satisfying the following conditions:

\begin{align}
V(\Gamma'_1)=V(\Gamma_1),\;\;\; \text{and} \label{replacement1}\\
\text{$Orb_{\mathbb{Z}}(\Gamma'_1)$ decomposes $K_U$.}\label{replacement2}
\end{align}

Indeed, letting $\Gamma' = (\Gamma\setminus\Gamma_1)\ \cup\  \Gamma'_1$,  by condition (<ref>) we have that $\Gamma'$ is a $k$-star with vertex-set $V$.
Recalling that $Orb_{\mathbb{Z}}(\Gamma_1)$ decomposes $K_U$,
by condition (<ref>) it follows that
$Orb_{\mathbb{Z}}(\Gamma')$ and $Orb_{\mathbb{Z}}(\Gamma)$ decompose the same graph, that is, $K_V$.
Hence, $Orb_{\mathbb{Z}}(\Gamma')$  is a factorization of $K_V$ into $k$-stars.

Let $k=h+3$ with $h\geq 2$.
It is left to construct an $h$-star $\Gamma'_{1,h}$ satisfying conditions
(<ref>) and (<ref>), for every $h\geq 2$.
For sake of clarity, in the rest of the proof we identify $U=\mathbb{Z}\times\{0\}$ with $\mathbb{Z}$.
Therefore, $\Gamma_1$ is the $1$-star centered in $0$ with
$\Gamma_1(0)=\{i: i\geq 1\}$.

Let $\Delta_j$ and $\Delta^*_{j}$ be the $1$-stars centered in $c_j=2(2^j-1)$
such that

\begin{align*}
&  \Delta_j(c_j) = \{c_j+i : 0 < i\equiv 2^j \pmod{2^{j+1}}\},\\
& \Delta^*_j(c_j) = \{c_j+i : 0 < i\equiv 0 \pmod{2^{j}}\},
\end{align*}

for $j\geq 0$,
and set
$\Gamma'_{1,h} = \Delta_0\ \cup\  \Delta_1\ \cup \ldots \cup\ \Delta_{h-2}\ \cup\ \Delta^*_{h-1}$ for $h\geq 2$.

It is not difficult to check that
$\{\Delta_j-c_j: 0\leq j\leq h-2\}\ \cup\ \{\Delta^*_{h-1} - c_{h-1}\}$ decomposes $\Gamma_1$.
Therefore, $Orb_{\mathbb{Z}}(\Gamma'_{1,h})$ and $Orb_{\mathbb{Z}}(\Gamma_1)$ decompose the same graph, that is, $K_U$. Hence, $\Gamma'_{1,h}$ satisfies condition (<ref>).

We show that $\Gamma'_{1,h}$ is an $h$-star satisfying condition (<ref>) by induction on $h$.
If $h=2$, then $V(\Delta_1)=\{0,1,3,5,\ldots\}$ and $V(\Delta_2)=\{2,4,6,\ldots\}$
Therefore, $\Gamma'_{1,2} = \Delta_0\ \cup\ \Delta^*_1$
is a $2$-star with the same vertex-set as  $\Gamma_1$.
Now assume that $\Gamma'_{1,h}$ is an $h$-star  satisfying condition (<ref>)
for some $h\geq 2$. Recalling the definition of $\Gamma'_{1,h}$ and $\Gamma'_{1,h+1}$,
and considering that the vertex-sets of $\Delta_{h-1}$ and $\Delta^*_{h}$ partition
$V(\Delta^*_{h-1})$, we have that $\Gamma'_{1,h+1}$ is an $(h+1)$-star with the same
vertex-set as $\Gamma'_{1,h}$, that is, $V(\Gamma_1)$, and this concludes the proof.

Propositions  <ref>, <ref> and <ref> leave open $FP(S_k)$ only when $k=3$.
In this case, an approach similar to Theorem <ref> cannot work, as shown in the following.

There is no $3$-star $\Gamma$
with vertex-set $V=\mathbb{Z}\times \{0,1, \ldots,k\}$ such that
the $\mathbb{Z}$-orbit of $\Gamma$ is an $S_3$-factorization of $K_V$

Assume for a contradiction that there exists a
$3$-star $\Gamma$ with vertex-set $V=\mathbb{Z}\times \{0,1, \ldots,k\}$ such that
$\mathcal{G} = Orb_{\mathbb{Z}}(\Gamma)$ is a factorization of $K_V$.

We first notice that $\Gamma$ must have at least a center in $\mathbb{Z}\times \{i\}$, for every
$i\in\{0,1, \ldots,k\}$. Indeed, if $\Gamma$ has no center in $\mathbb{Z}\times \{i\}$ for some $i\in\{0,1, \ldots,k\}$, then no edge of $K_{\mathbb{Z}\times \{i\}}$ can be covered by $\mathcal{G}$. Since $\Gamma$ has $3$ centers, it follows that $k\leq 2$.
Note that if $k=2$, the centers of $\Gamma$ must be $x_0, y_1, z_2$ for some $x,y,z\in\mathbb{Z}$, but in this case the edge $\{x_0, y_1\}$ cannot lie in any translate of $\Gamma$.
Therefore $k\leq 1$.

If $k=1$, without loss of generality we can assume that the centers of $\Gamma$
are $0_0$, $x_1$ and $y_1$ with $x\not=y$. Since the edge $\{0_0,x_1\}$ does not belong to $\Gamma$,
it lies in some of its translates, say $\Gamma+z$ with $z\neq 0$. This is equivalent to saying that
$\{(-z)_0,(x-z)_1\}\in \Gamma$. It follows that $x-z=y$, hence $\{(y-x)_0,y_1\}\in \Gamma$.
Similarly, we can show that $\{(x-y)_0,x_1\}\in \Gamma$. It follows that $\Gamma$ cannot contain
the edges $\{0_0, (x-y)_0\}$ and $\{0_0, (y-x)_0\}$. This implies that no edge of the form
$\{w_0, (x-y+w)_0\}$ lie in any translate of $\Gamma$, contradicting again the assumption that
$\mathcal{G}$ is a factorization of $K_V$. Therefore $k=0$.

Let $V=\Z$ and denote by
$\Delta \Gamma$ the multiset of all differences $y-x$ between any two adjcent vertices $x$ and $y$ of $\Gamma$, with $x<y$:

\[
\Delta \Gamma = \{y-x: \{x,y\}\in E(\Gamma),\; x<y\}.
\]

It is not difficult to see that $\mathcal{G} = Orb_{\mathbb{Z}}(\Gamma)$ is a factorization of $K_\Z$ if and only if $\Delta \Gamma = \mathbb{N}\setminus\{0\}$.
Denoting by $\Gamma+i$ the translate of $\Gamma$ obtained by replacing each vertex
$x\in V(\Gamma)$  with $x+i$, one can easily see that $\Delta (\Gamma+i) = \Delta \Gamma$ for every $i\in\Z$.
Therefore, up to a translation, we can assume that
the centers of $\Gamma$ are $0, x, n$ with $0<x<n$. Now, for every $i\geq n$,
denote by $\Gamma_i$ the induce subgraph of $\Gamma$ with vertex-set $\{0,1, \ldots,i\}$.
Also, let $\Gamma^*$ be the induced subgraph of $\Gamma$ on the vertices
$\{-3,-2,-1, 0, x, n\}$.
$|\Delta \Gamma^*|=3$, $|\Delta \Gamma_i| = i-2$ and $\Delta \Gamma_i \subset \{1,2,\ldots, i\}$. Also,
since the multiset $\Delta \Gamma$ contains all positive integers with no repetition,
it follows that $\Delta \Gamma^*$ and $\Delta \Gamma_i$ are disjoint, hence
$\Delta \Gamma_i \subset \{1,2,\ldots, i\}\setminus \Delta \Gamma^*$ for every $i\geq n$.
Then, for $i=max (\Delta \Gamma^*)$, we obtain the following contradiction:
$i-2 = |\Delta \Gamma_i| \leq |\{1,2,\ldots, i\}\setminus \Delta \Gamma^*| = i-3$.

§ THE RESOLVABILITY PROBLEM

Theorem <ref> allows us to construct decompositions of $K_\aleph$ into $\aleph$ graphs of specified type.
More precisely, we have the following.

Let $\mathcal{F}=\{F_{\alpha}: \alpha\in \mathcal{A}\}$ be an infinite family of (non-empty)
$\aleph$-bounded graphs,
where $\aleph=|\mathcal{A}|$.
Then there exists a decomposition
$\mathcal{G}=\{\Gamma_{\alpha}: \alpha\in \mathcal{A}\}$ of $K_{\aleph}$ such that
each $\Gamma_{\alpha}$ is isomorphic to $F_{\alpha}$.

Furthermore, if the domination number of some graph $F_\beta$ is less than $\aleph$, then
$|V(K_\aleph) \setminus V(\Gamma_\beta)|= \aleph$.
Otherwise, for every $0\preceq \aleph' \preceq \aleph$,
the decomposition $\mathcal{G}$ can be constructed so that
$|V(K_\aleph) \setminus V(\Gamma_\beta)|= \aleph'$.

For every $\alpha\in\mathcal{A}$, set $\aleph_{\alpha}=\aleph$ if the domination number of $F_\alpha$ is less than $\aleph$; otherwise, let $0\preceq \aleph_\alpha \preceq \aleph$.
By adding to each graph $F_\alpha$ a set of $\aleph_\alpha$  isolated vertices
we obtain a graph $F'_\alpha$ whose order and domination number are $\aleph$.
Since the assumptions of Theorem <ref> are satisfied, there exists a factorization
$\mathcal{G'}=\{\Gamma'_{\alpha}: \alpha\in \mathcal{A}\}$ of $K_{\aleph}$ such that
each $\Gamma'_{\alpha}$ is isomorphic to $F'_{\alpha}$. By replacing $\Gamma'_\alpha$ with the isomorphic copy of $F_{\alpha}$, we obtain the desired decomposition $\mathcal{G}$.

Inspired by [9], we ask under which conditions a decomposition $\mathcal{G}$ of $K_{\aleph}$
is resolvable, namely, its graphs can be partitioned into factors of $K_{\aleph}$, also called
resolution classes. It follows that a resolvable decomposition $\mathcal{G}$ of $K_{\aleph}$
must satisfy the following two conditions:

if $\Gamma\in\mathcal{G}$ is not a factor of $K_{\aleph}$,
then  $|V(K_{\aleph})\setminus V(\Gamma)|\geq \min\{|\Gamma|: \Gamma\in \mathcal{G}\}$.

for every $x,y,z\in V(K_\aleph)$,

\[\mathcal{G}(z) \subseteq \mathcal{G}(x) \cup \mathcal{G}(y)
\;\;\Rightarrow\;\;
\mathcal{G}(z) \supseteq \mathcal{G}(x) \cap \mathcal{G}(y),
\]

where $\mathcal{G}(v)=\{\Gamma\in \mathcal{G}: v\in V(\Gamma)\}$ is the set of all graphs of
$\mathcal{G}$ passing through $v$.

In the following, we easily construct decompositions of $K_\aleph$ that do not satisfy the above conditions, and therefore they are non-resolvable.

Let $\mathcal{F}=\{F_{\alpha}: \alpha\in \mathcal{A}\}$ be an infinite family of (non-empty)
$\aleph$-bounded graphs,
where $\aleph=|\mathcal{A}|$. Also, assume that the domination number of at least one of its graphs,
say $F_\beta$, is $\aleph$.
Then, by applying Corollary <ref>
with $\aleph'\prec \min\{|\Gamma_\alpha|: \alpha\in \mathcal{A}\}$, we construct a decomposition that does not satisfy condition N1.

For instance, if $\aleph=|\mathbb{N}|$, each $F_{\alpha}$ is a
countable locally finite graph
(hence, its domination number is $\aleph$) and $\aleph'=1$ for every $\beta \in \mathbb{N}$, then
we construct a decomposition
$\mathcal{G}=\{G_\beta: \beta\in \mathbb{N}\}$ of $K_{\mathbb{N}}$ into connected regular graphs where $V(G_\beta)=\mathbb{N}\setminus \{x_\beta\}$ for some $x_\beta\in \mathbb{N}$. Clearly, no graph of
$\mathcal{G}$ is a factor  of $K_{\mathbb{N}}$, and any two graphs of $\mathcal{G}$ have common vertices. Therefore, $\mathcal{G}$ is not resolvable.

Let $\mathcal{G}$ be any decomposition of the infinite complete graph $K_{V}$ (for example, one of those constructed by Corollary <ref>).
Let $y$ and $z$ be vertices not belonging to $K_{V}$ and
set $W=V \ \cup\ \{y,z\}$. We can easily extend $\mathcal{G}$ to a non-resolvable decomposition $\mathcal{G'}$ of $K_W$ in the following way.

Choose $x\in V$ and let $\mathcal{C}$ be the following family of paths of length 1 or 2:

\[
\mathcal{C}=\{[y,v,z]: v\in V\setminus\{x\}\}\ \cup\ \{[x,z,y], [x,y]\}.
\]

Clearly, $\mathcal{C}$ decomposes $K_W \setminus K_V$,
hence $\mathcal{G}' = \mathcal{G}\ \cup\ \mathcal{C}$ is a decomposition of $K_{W}$.
Also, $x,y$ and $z$ do not satisfy condition N2, since
$\mathcal{G}'(z)\subseteq \mathcal{G}'(x)\ \cup\ \mathcal{G}'(y)$,
while $[x,y]$ belongs to $\mathcal{G}'(x)\cap \mathcal{G}'(y)$, but not to $\mathcal{G}'(z)$.
Therefore, $\mathcal{G}'$ is non-resolvable.  Indeed, any resolution class of $\mathcal{G}'$
could cover the vertex  $z$ only with graphs passing through $x$ or $y$. This means that
the graph $[x,y]$ cannot belong to any resolution class of $\mathcal{G}'$.

The following result provides sufficient conditions for a decomposition $\mathcal{G}$ to be resolvable.

Let $\mathcal{G}$ be a decomposition of the infinite complete graph $K_{\aleph}$ satisfying the following properties for some $\aleph'\prec \aleph$:

R1. each graph in $\mathcal{G}$ is $\aleph'$-bounded;

R2. $|\mathcal{G}(x)\ \cap \ \mathcal{G}(y)|\preceq\aleph'$
for every distinct $x,y\in V(K_{\aleph})$.

Then $\mathcal{G}$ is resolvable.

Let $\mathcal{G}=\{G_{\alpha}: \alpha\in \mathcal{A}\}$.
We consider a well-order $\prec$ on $\mathcal{A}$
satisfying Corollary <ref>.
Since the graphs of $\mathcal{G}$ are $\aleph'$-bounded, we have that $|\mathcal{A}|=\aleph$ and we can assume $V(K_{\aleph})= \mathcal{A}$.
Here we need to construct an ascending chain $(\mathcal{G}_{\gamma})_{\gamma\in \mathcal{A}}$
of families
$\mathcal{G}_{\gamma}:=\{\Gamma_{\alpha}^{\gamma}: \alpha\in \mathcal{A}_{\preceq \gamma}\}$
(where $\Gamma_{\alpha}^{\gamma'}$ is a subgraph of $\Gamma_{\alpha}^{\gamma}$ whenever
$\gamma'\preceq \gamma$) that satisfy the following proprieties:

($1_{\gamma}$) each $\Gamma_{\alpha}^{\gamma}$ is a vertex-disjoint union of graphs of

($2_{\gamma}$) for every $\alpha \in \mathcal{A}_{\preceq \gamma}$, $\gamma\in V(\Gamma_{\alpha}^{\gamma})$;

($3_{\gamma}$) $G_\gamma$ is contained in exactly one $\Gamma_{\alpha}^{\gamma}$ where $\alpha\in \mathcal{A}_{\preceq \gamma}$;

($4_{\gamma}$) for every $\alpha \in \mathcal{A}_{\preceq \gamma}$, $\Gamma_{\alpha}^{\gamma}$ is either a finite graph or  $(\aleph'\cdot|\mathcal{A}_{\preceq \gamma}|)$-bounded.

The desired resolution of $K_\aleph$ is then
$\mathcal{R}=\{\Gamma_\alpha: \alpha\in \mathcal{A}\}$,
where $\Gamma_\alpha =\bigcup_{\gamma\in\mathcal{A}} \Gamma^\gamma_\alpha$
for every $\alpha\in\mathcal{A}$.
Indeed, due to properties ($1_\gamma$) and ($2_\gamma$), each $\Gamma_{\alpha}$ is a resolution class of $\mathcal{G}$ and, by property ($3_\gamma$), $\mathcal{R}$ is a partition of $\mathcal{G}$ into resolution classes.

We proceed by transfinite induction on $\gamma$.

BASE CASE. Let $0=\min  X$.
By condition $R2$, if $0$ is not a vertex of $G_0$, $|\mathcal{G}(0)\cap \mathcal{G}(x)|\preceq \aleph'$ for any $x\in V(G_0)$. Since, due to condition $R1$, $|\mathcal{G}(0)|=\aleph$, there exists $G\in \mathcal{G}(0)$ disjoint from $G_0$.
Therefore we can define $\mathcal{G}_0=\{\Gamma_{0}^{0}\}$ where $\Gamma_{0}^{0}$ is either $G_0\cup G$ or, if $0$ belongs to $V(G_0)$, $G_0$.

TRANSFINITE INDUCTIVE STEP.
For every $\gamma'\prec \gamma$, we assume there is a family
$\mathcal{G}_{\gamma'}$ satisfying  $(i_{\gamma'})$ for $1\leq i\leq 4$.

We show that $\mathcal{G}_{\gamma'}$ can be extended to a family
$\mathcal{G}_{\gamma}$ that satisfies the same properties, $(i_{\gamma})$ for $1\leq i\leq 4$.

We are going to define, recursively, the graphs $\Gamma_{\alpha}^{\gamma}$ whenever $\alpha\preceq \gamma$.
First, we consider the case $\alpha\prec \gamma$. We start by setting $\Gamma_{\alpha}^{\prec \gamma}:=\bigcup_{\gamma'\prec\gamma} \Gamma_{\alpha}^{\gamma'}$.
Note that  property ($4_{\gamma'}$) guarantees that $\Gamma_{\alpha}^{\prec \gamma}$ is either finite or $|\Gamma_{\alpha}^{\prec\gamma}|\preceq\aleph'\cdot |\mathcal{A}_{\preceq\gamma}|$;
hence, $\Gamma_{\alpha}^{\prec \gamma}$ is $\aleph$-small.

* Base case.
If $\gamma\in V(\Gamma_{0}^{\prec\gamma})$, set $\Gamma_{0}^{\gamma}=\Gamma_{0}^{\prec\gamma}$.

If $\gamma\not \in V(\Gamma_{0}^{\prec\gamma})$, by condition R2 we have
$|\mathcal{G}(\gamma)\cap \mathcal{G}(x)|\preceq\aleph'$
for every $x\in V(\Gamma_0^{\prec\gamma})$.
Since $\Gamma_0^{\prec\gamma}$ is $\aleph$-small, this means that the family of graphs of $\mathcal{G}(\gamma)$ that intersect $V(\Gamma_0^{\prec\gamma})$ is $\aleph$-small.

Moreover, any $\Gamma_{\alpha}^{\prec \gamma}$ is either finite or
$(\aleph'\cdot |\mathcal{A}_{\preceq\gamma}|)$-bounded
(note that $\aleph'\cdot |\mathcal{A}_{\preceq\gamma}| \prec \aleph$,
since $|\mathcal{A}_{\preceq\gamma}|\prec \aleph$).
Hence, the set of graphs in $\mathcal{G}(\gamma)$ that are contained in some $\Gamma_{\alpha}^{\prec \gamma}$ is $\aleph$-small.

Finally, by condition R1, we have that $|\mathcal{G}(\gamma)|=\aleph$.
Therefore, there exists a graph $G\in \mathcal{G}(\gamma)$ that is not contained in any $\Gamma_{\alpha}^{\prec \gamma}$ and such that $V(G)\cap V(\Gamma_0^{\prec\gamma})=\emptyset$.
Then, we set $\Gamma_{0}^{\gamma}=\Gamma_0^{\prec\gamma}\cup G$.

* Inductive step.
Let $\alpha\prec \gamma$.
If $\gamma\in V(\Gamma_{\alpha}^{\prec\gamma})$,
set $\Gamma_{\alpha}^{\gamma}=\Gamma_{\alpha}^{\prec\gamma}$.
Otherwise, by proceeding as in the previous case, we obtain the existence
of a graph $G\in \mathcal{G}(\gamma)$ that is not in any $\Gamma_{\alpha'}^{\prec \gamma}$
or any $\Gamma_{\alpha''}^{\gamma}$ (where $\alpha'\prec \gamma$ and $\alpha''\prec \alpha$),
and  such that $V(G)\cap V(\Gamma_{\alpha}^{\prec\gamma})=\emptyset$.
In this case, we  set $\Gamma_{\alpha}^{\gamma}=\Gamma_{\alpha}^{\prec\gamma}\cup G$.

It is left to define $\Gamma_{\gamma}^{\gamma}$. We proceed by constructing,
recursively, an ascending chain of graphs $\Gamma_{\gamma}^{\alpha}$, for $\alpha\in \mathcal{A}_{\preceq \gamma}$, that are either finite or $(\aleph'\cdot|\mathcal{A}_{\preceq\gamma}|)$-bounded.

* Base case. Let us first suppose that $G_{\gamma}$ is not contained in any $\Gamma_{\alpha'}^{\gamma}$ (where $\alpha'\prec\gamma$).
Again, by conditions R1 and R2, there exists $G\in \mathcal{G}(0)$ that is also not contained in any $\Gamma_{\alpha'}^{\gamma}$ such that $G$ is either $G_{\gamma}$ or is disjoint from $G_{\gamma}$. We set $\Gamma_\gamma^0$ to be $G_{\gamma}\cup G$.
Otherwise, we set $\Gamma_\gamma^0$ to be any graph $G$ in $\mathcal{G}(0)$ that is  not contained in any $\Gamma_{\alpha'}^{\gamma}$.

* Inductive step. Let us suppose that $\alpha\not=0$ and that we have defined $\Gamma_{\gamma}^{\alpha'}$ for every $\alpha'\prec \alpha$. Here we set $\Gamma_{\gamma}^{\prec \alpha}$ to be $\bigcup_{\alpha'\prec \alpha}  \Gamma_{\gamma}^{\alpha'}$. Note that, for construction, $\Gamma_{\gamma}^{\prec \alpha}$ is either a finite graph or $|\Gamma_{\gamma}^{\prec \alpha}|\preceq\aleph'\cdot |\mathcal{A}_{\preceq\gamma}|$.
If $\alpha$ belongs to $V(\Gamma_{\gamma}^{\prec \alpha})$, we set $\Gamma_{\gamma}^{\alpha}$ to be $\Gamma_{\gamma}^{\prec \alpha}$.
Otherwise, proceeding as in the previous case, we obtain that there exists $G\in \mathcal{G}(\alpha)$ disjoint from $\Gamma_{\gamma}^{\prec \alpha}$ that does not belong to any of the $\Gamma_{\alpha'}^{\gamma}$. Now we set $\Gamma_{\gamma}^{\alpha}$ to be $G\cup \Gamma_{\gamma}^{\prec \alpha}$.

Then the family $\mathcal{G}_{\gamma}=\{\Gamma_{\alpha}^{\gamma}: \alpha\in \mathcal{A}_{\preceq \gamma}\}$ satisfies the properties $(1_{\gamma})$, $(2_{\gamma})$, $(3_{\gamma})$ and $(4_{\gamma})$ for construction.

A cardinal $\aleph$ is said to be regular if any $\aleph$-small union of $\aleph$-small sets (resp. graphs) is still an $\aleph$-small set (resp. graph) otherwise it is said to be singular.
It is easy to see that, for regular cardinals, conditions $R1$ and $R2$ of Theorem <ref> can be relaxed to:

$R1'.$ each graph in $\mathcal{G}$ is $\aleph$-small;

$R2'.$ $|\mathcal{G}(x)\ \cap \ \mathcal{G}(y)|\prec\aleph$
for every distinct $x,y\in V(K_{\aleph})$.

However, if $\aleph$ is a singular cardinal, then conditions $R1'$ and $R2'$ are no longer sufficient.
Indeed, we can construct a decomposition $\mathcal{G}$ of $K_{\aleph}$ into $\aleph$-small graphs such that

a. $|\mathcal{G}|$ is $\aleph$-small,

b. $\mathcal{G}$ satisfies conditions $R1'$ and $R2'$,

c. there are two (possibly isolated) vertices $x$ and $y$ belonging to every graphs of $\mathcal{G}$, that is, $\mathcal{G}=\mathcal{G}(x) \cap \mathcal{G}(y)$

Then, choosing any vertex $z$ such that $\mathcal{G}(z)\not=\mathcal{G}$, we have that

\[\mathcal{G}(z) \subseteq \mathcal{G}(x) \cup \mathcal{G}(y)=\mathcal{G}
\;\;\ \mbox{ but }\;\;
\mathcal{G}(z) \not\supseteq \mathcal{G}(x) \cap \mathcal{G}(y)=\mathcal{G}.
\]

This means that condition N2 does not hold, therefore the decomposition $\mathcal{G}$ is not resolvable.

We conclude by showing that there is always a resolution for an `almost' $2$-design with
blocks that are $\aleph'$-bounded for some $\aleph'\prec \aleph$,
that is, a decomposition of $K_{\aleph}$ whose graphs are almost all  $\aleph'$-bounded complete graphs.
This extends some results on the resolvability of $2$-designs given in [9].

Let $\mathcal{G}$ be a decomposition of the infinite complete graph $K_\aleph$ into $\aleph'$-bounded graphs for some $\aleph'\prec \aleph$, where $\aleph'$ is not necessarily infinite.
If the subset of $\mathcal{G}$ consisting of all non-complete graphs is $\aleph'$-bounded,
then $\mathcal{G}$ has a resolution.

By assumption, condition R1 of Theorem <ref> holds. To prove that $\mathcal{G}$ satisfies condition R2 for some $\aleph''\prec \aleph$, we assume for a contradiction the existence of vertices $x$ and $y$ such that $|\mathcal{G}(x)\ \cap \ \mathcal{G}(y)|\succ \aleph'':=(\aleph'+1)$. It follows that there are at least two complete graphs in $\mathcal{G}(x)\ \cap \ \mathcal{G}(y)$, meaning that the edge $\{x,y\}$ is covered more than once by graphs in $\mathcal{G}$, and this is a contradiction. The assertion follows from Theorem <ref>.

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