# Decorrelation of participant and spectator angular momenta in heavy-ion
collisions

Joseph R. Adams, Michael A. Lisa  Department of Physics, Ohio State
University, Columbus, Ohio 43210 USA

###### Abstract

High-energy heavy-ion collisions contain enormous angular momentum,
$|\vec{J}|$, which is $\mathcal{O}(10^{3}-10^{6}\hbar)$ in the range of
collision energy, $\sqrt{s_{\mathrm{NN}}}$, spanned experimentally by the
Relativistic Heavy Ion Collider (RHIC) and the Large Hadron Collider (LHC). A
fraction of $\vec{J}$ is transferred to the overlapping collision region,
which is indispensable for measuring observables such as vorticity-driven
hadron spin alignment with $\hat{J}$. Experiments estimate the orientation of
$\hat{J}$ of the participant nucleons within the collision overlap region,
${\hat{J}_{\mathrm{part}}}$, by using that of the forward- and backward-going
spectating nucleons ${\hat{J}_{\mathrm{spec}}}$. Using two models, we study
the decorrelation between ${\hat{J}_{\mathrm{part}}}$ and
${\hat{J}_{\mathrm{spec}}}$, driven both by angular-momentum conservation and
event-by-event fluctuations, as well as by the decorrelation between the
orientation of the elliptic overlap region and the
${\hat{J}_{\mathrm{part}}}$. $\sqrt{s_{\mathrm{NN}}}$-dependent decorrelation
is observed in both of these cases and is large enough to be an important
corrective factor used when experimentally observing phenomena driven by
$\vec{J}$.

##  I Introduction

Relativistic heavy-ion collisions generate sufficient energy densities to
momentarily [$\mathcal{O}$(1 fm/$c$)] deconfine constituent quarks, and
thereby create the conditions required to study the strong nuclear force
described by quantum chromodynamics (QCD). The state of matter produced is the
so-called quark-gluon plasma (QGP) Shuryak (1980); Adams  _et al._ (2005);
Adcox  _et al._ (2005); Back  _et al._ (2005); Arsene  _et al._ (2005), which
has been the subject of intense study for decades; theorists and
experimentalists study a variety of phenomena over a wide range of center-of-
mass, nucleon-nucleon collision energy, $\sqrt{s_{\mathrm{NN}}}$, in an effort
to characterize the QCD phase diagram Akiba  _et al._ (2015).

Phenomena driven by the angular momentum of participating nucleons within the
collision overlap region, ${\vec{J}_{\mathrm{part}}}$, have been of much
interest Becattini and Lisa (2020); Liang and Wang (2005); Becattini and
Piccinini (2008); Betz  _et al._ (2007); Abelev  _et al._ (2007); Adamczyk
_et al._ (2017); Acharya  _et al._ (2020); Adam  _et al._ (2021); Jiang  _et
al._ (2016). Experimentally, the orientation of ${\hat{J}_{\mathrm{part}}}$ is
not known exactly and must be approximated with the orientation of the angular
momentum of the forward- and backward-going “spectator” nucleons,
${\hat{J}_{\mathrm{spec}}}$, which are characterized by having a “forward”
rapidity, $y$, or pseudo-rapidity, $\eta$, with $|y|,|\eta|\gtrsim 1-2$. The
QGP state, formed in the collision overlap region, ultimately emits particles
across a range of rapidity, but typically only the mid-rapidity
($|y|,|\eta|\lesssim 1-2$) particles’ paths are able to be reconstructed
experimentally. Observables such as the spin orientations of hadrons,
$\vec{S}_{\rm H}$, which have been shown to be globally aligned with
${\hat{J}_{\mathrm{syst}}}$ Adamczyk  _et al._ (2017); Acharya  _et al._
(2020); Adam  _et al._ (2021); Abdallah  _et al._ (2021), are therefore only
accessible at mid-rapidity. Ideally, then, the correlation between observables
such as $\vec{S}_{\rm H}$ and the angular-momentum orientation of the
corresponding mid-rapidity region, ${\hat{J}_{\mathrm{mid}}}$, could be
studied.

The angular momentum of the system, ${\vec{J}_{\mathrm{syst}}}$, is
perpendicular in the transverse plane (which is orthogonal to the beam axis)
to the the impact parameter, $\vec{b}$, connecting the centers of the two
nuclei. ${\hat{J}_{\mathrm{part}}}$ instead fluctuates about
${\hat{J}_{\mathrm{syst}}}$ on an event-by-event basis, due to the randomness
of nucleon positions within the colliding nuclei. While the effects of
initial-state fluctuations in heavy-ion collisions have been thoroughly
investigatedAlver and Roland (2010); Bzdak and Skokov (2012); Bloczynski  _et
al._ (2013); Vovchenko  _et al._ (2013); Qin  _et al._ (2010); Lacey  _et al._
(2011); Qian and Heinz (2016); Jia (2014); Luzum and Petersen (2014); Magdy
_et al._ (2020); Gardim  _et al._ (2021); Bhalerao  _et al._ (2019); Ma  _et
al._ (2016), it has yet to be discussed in the context of decorrelation
between ${\hat{J}_{\mathrm{part}}}$ and ${\hat{J}_{\mathrm{spec}}}$.

Through the use of a simple and intuitive model, we show in this study that
the effects of both initial-state fluctuations and angular-momentum
conservation lead to a significant decorrelation between the true
${\hat{J}_{\mathrm{part}}}$ and ${\hat{J}_{\mathrm{part}}}$ as estimated by
${\hat{J}_{\mathrm{spec}}}$. Furthermore, by using a more realistic model
which evolves in time and simulates parton interactions, we find a
significantly larger suppression of the correlation between
${\hat{J}_{\mathrm{spec}}}$ and the ${\hat{J}_{\mathrm{mid}}}$ that is
experimentally of interest. This decorrelation is strongly dependent on
$\sqrt{s_{\mathrm{NN}}}$ and would require experimental observations of
phenomena driven by angular momentum to correct for this effect. While it may
seem natural to attempt to avoid such corrective factors by measuring
${\hat{J}_{\mathrm{mid}}}$ directly from the azimuthal distribution of
particles emitted at mid-rapidity, we find a decorrelation between the
orientation of the roughly elliptic shape of the overlap region, practically
disallowing such a method.

##  II Models

The first of two models used to study these effects is a simple Monte-Carlo
Glauber (MCG) model Miller  _et al._ (2007). Our MCG model consists of
randomly generating angular and radial coordinates according to the Woods-
Saxon distribution,

$\ \rho(r)=\frac{\rho_{0}}{1+e^{\frac{r-R}{a}}},$ (1)

with the appropriate Jacobians. In this study, we look at ${}^{197}_{97}$Au,
where $\rho_{0}=0.1693$ fm-3, $R=6.38$ fm, and $a=0.535$ fm Fricke and Heilig
. An impact parameter $|\vec{b}|$ is chosen according to $dN/db\propto b$, and
two nuclei are generated around two points a distance $b_{i}$ from each other
before re-centering the nuclei to maintain the chosen $b$. While generating
nucleon positions, any newly generated nucleon whose center lies within 0.9 fm
of another nucleon’s center within the same nucleus is regenerated.

Figure 1:  A series of collisions generated using our MCG model at varying b,
viewed in the transverse plane. The participants are outlined in black and the
elliptic fit to their positions is displayed in dashed yellow with a line
through the major axis. The events shown are typical, in that
${\hat{J}_{\mathrm{part}}}\cdot{\hat{J}_{\mathrm{spec}}}|_{b}\approx\left<{\hat{J}_{\mathrm{part}}}\cdot{\hat{J}_{\mathrm{spec}}}\right>\Big{\rvert}_{b}$
and
$|\sin(\Phi_{2})|\Big{\rvert}_{b}\approx\left<|\sin(\Phi_{2})|\right>\Big{\rvert}_{b}$.
Figure 2:  $|{\vec{J}_{\mathrm{part}}}|$ is the largest at $b\approx 3.5$ fm,
but is sizable for all but peripheral collisions. For very central collisions
($b\lesssim 0.5$ fm), $|{\vec{J}_{\mathrm{part}}}|$ and
$|{\vec{J}_{\mathrm{spec}}}|$ each become larger than
$|{\vec{J}_{\mathrm{syst}}}|$ since spectators still exist while
$|{\vec{J}_{\mathrm{syst}}}|\rightarrow 0$ as $b\rightarrow 0$.

The qualification that a nucleon must satisfy to be considered a participant
is that its center must lie within $d_{\perp}$ of at least one nucleon from
the other nucleus. $d_{\perp}$ is related to the beam energy and the nucleon-
nucleon inelastic cross section at that energy, which is parameterized
according to Montanet  _et al._ (1994):

$\ \sigma^{\mathrm{total}}_{\mathrm{NN}}=48+0.522(\ln{p})^{2}-4.51\ln{p}\\\
\sigma_{\mathrm{NN}}^{\mathrm{elastic}}=11.9+26.9p^{-1.21}+0.169(\ln{p})^{2}-1.85\ln{p}\\\
d_{\perp}=\sqrt{\frac{\sigma_{\mathrm{NN}}^{\mathrm{total}}-\sigma_{\mathrm{NN}}^{\mathrm{elastic}}}{\pi}}=\sqrt{\frac{\sigma_{\mathrm{NN}}^{\mathrm{inelastic}}}{\pi}},$
(2)

where $p$ is the center-of-mass nucleon momentum. The dependence of
$d_{\perp}$ on $\sqrt{s_{\mathrm{NN}}}$ is weak. For the collision energy
$\sqrt{s_{\mathrm{{NN}}}}=27$ GeV, somewhat arbitrarily chosen for our
calculations, $d_{\perp}=0.984$ fm.

A series of typical peripheral collisions using the MCG model at varying $b$
is shown in Fig. 1, where $\Phi_{2}$ describes the orientation of the
collision and is defined in Eq. 4. The directions of
${\hat{J}_{\mathrm{syst}}}$, shown as magenta (dark gray) arrows,
${\hat{J}_{\mathrm{part}}}$, shown as the gray (medium gray) arrows, and
${\hat{J}_{\mathrm{spec}}}$, shown as the green (light gray) arrows, are shown
for each collision. ${\hat{J}_{\mathrm{syst}}}$ always points in the
$-\hat{y}$ direction while ${\hat{J}_{\mathrm{part}}}$ and
${\hat{J}_{\mathrm{spec}}}$ fluctuate about the $-\hat{y}$ direction and point
on opposite sides of ${\hat{J}_{\mathrm{syst}}}$. While our MCG model
calculations serve as a nice baseline for building intuitions, they do not
incorporate any time evolution of the system and therefore do not allow us to
study the effects of angular-momentum redistribution through particle
interactions or to select regions in rapidity.

As angular-momentum-driven phenomena are interested mainly in the QGP phase,
we do not want to concern ourselves with late-stage interactions or decays
which will act with a disproportionately large lever arm on the angular
momentum of the region. The string-melting version of the a multi-phase
transport” (AMPT) model Lin  _et al._ (2005) provides the ideal environment
for a study of these angular momentum correlations with a more realistic
description, while still allowing the user to ignore late-stage interactions
and decays. The user has access to the positions and momenta of the spectators
and of the partons at hadronization. Although hadronization will redistribute
angular momentum to some degree, this is a sub-dominant effect.

AMPT uses the heavy-ion jet interaction generator (HIJING) Gyulassy and Wang
(1994) for initial conditions and Zhang’s parton cascade (ZPC) Zhang (1998)
for handling partonic interactions. The Lund string fragmentation model is
used for hadronization and a relativistic transport (ART) model is used for
treating hadronic scatterings. For angular momentum calculations, we are only
interested in the state of partons at the moment of hadronization, which is at
the end of the ZPC stage.

The input parameters to AMPT, besides $\sqrt{s_{\mathrm{NN}}}$, the range of
$b$, and the number of collisions, are not changed. For MCG and AMPT, 50K
events are generated with $0\leq b\leq 14$ fm. Only one collision energy is
studied in MCG as the only energy-dependent effect is a slight reduction in
$r_{\perp}$ with $\sqrt{s_{\mathrm{NN}}}$; the somewhat arbitrary choice is
$\sqrt{s_{\mathrm{NN}}}=27$ GeV.

##  III ${\vec{J}_{\mathrm{part}}},{\vec{J}_{\mathrm{mid}}}$ Correlations with
${\hat{J}_{\mathrm{syst}}},{\hat{J}_{\mathrm{spec}}}$

Figure 3: ${\hat{J}_{\mathrm{part}}}$ and ${\hat{J}_{\mathrm{syst}}}$ are
poorly correlated for central and peripheral collisions but are well
correlated in between. The correlation between ${\hat{J}_{\mathrm{part}}}$ and
${\hat{J}_{\mathrm{spec}}}$ is smaller than the correlation between
${\hat{J}_{\mathrm{part}}}$ and ${\hat{J}_{\mathrm{syst}}}$, an effect driven
by conservation of angular momentum. In AMPT, shown as the lower two lines, we
choose spectators with the somewhat arbitrary cut $|y|>2$ and in this figure
the remaining particles are considered participants.

In central heavy-ion collisions ($b\lesssim 3$ fm),
$|{\vec{J}_{\mathrm{syst}}}|$ and $|{\vec{J}_{\mathrm{part}}}|$ become smaller
as $b\rightarrow 0$ while $|{\vec{J}_{\mathrm{spec}}}|$ remains large; many
spectators still exist in these collisions (as seen for example in Fig. 1),
which carry a large lever arm. Because of this, $|{\vec{J}_{\mathrm{spec}}}|$
and $|{\vec{J}_{\mathrm{part}}}|$ are non-zero even as $b\rightarrow 0$ (see
Fig. 2). As $b\rightarrow 0$, ${\hat{J}_{\mathrm{spec}}}$ becomes more random
and so, therefore, does ${\hat{J}_{\mathrm{part}}}$ due to conservation of
angular momentum. In peripheral collisions ($b\gtrsim 9$ fm)
$|{\vec{J}_{\mathrm{spec}}}|$ dominates the contribution to
$|{\vec{J}_{\mathrm{syst}}}|$ but the effects of initial-state fluctuations on
${\hat{J}_{\mathrm{spec}}}$ diminish as the number of spectators increases, so
${\hat{J}_{\mathrm{spec}}}$ becomes well aligned with
${\vec{J}_{\mathrm{syst}}}$; however, the number of participants drops as does
the contribution of $|{\vec{J}_{\mathrm{part}}}|$ to
$|{\vec{J}_{\mathrm{syst}}}|$, so initial-state fluctuations play a
significant role in the orientation of ${\hat{J}_{\mathrm{part}}}$. We might
therefore expect ${\hat{J}_{\mathrm{part}}}$ and ${\hat{J}_{\mathrm{syst}}}$
to be poorly correlated in central and peripheral collisions. In mid-central
collisions, however, there are enough of both participants and spectators that
initial-state fluctuations play a small role in the orientations of
${\hat{J}_{\mathrm{part}}}$ and ${\hat{J}_{\mathrm{spec}}}$ and we might
therefore expect them to be well correlated in these collisions. We indeed see
this behavior in the solid lines in Fig. 3 measuring
${\hat{J}_{\mathrm{part}}}\cdot{\hat{J}_{\mathrm{syst}}}$ with the MCG and
AMPT models. Here and henceforth we use the rapidity cut $|y|>2$ in AMPT to
approximately isolate the spectators, as would be done experimentally. When
choosing the upper limit of $|y|=2$ to define the participant region in AMPT,
no particles are excluded and we therefore see quite good agreement between
the two models.

When instead measuring the correlation between the participants and the
spectators, we see that
${\hat{J}_{\mathrm{part}}}\cdot{\hat{J}_{\mathrm{spec}}}<{\hat{J}_{\mathrm{part}}}\cdot{\hat{J}_{\mathrm{syst}}}$;
this is true both on average as well as event by event, and must be so because
of conservation of angular momentum. This is represented in Fig. 1 as a
cartoon of mid-central collisions within the MCG model viewed in the
transverse plane. By design, ${\hat{J}_{\mathrm{syst}}}||-\hat{y}$ but
initial-state fluctuations will generate a deviation of
${\hat{J}_{\mathrm{part}}}$(${\hat{J}_{\mathrm{spec}}}$) from $-\hat{y}$ and
because of angular-momentum conservation
${\hat{J}_{\mathrm{spec}}}$(${\hat{J}_{\mathrm{part}}}$) must point along the
“other side” of $-\hat{y}$; i.e., the angle between
${\hat{J}_{\mathrm{part}}}$ and ${\hat{J}_{\mathrm{spec}}}$ must be larger
than the angle between ${\hat{J}_{\mathrm{part}}}$ and
${\hat{J}_{\mathrm{syst}}}$.

Experiments are typically set up to identify particles with tracking at mid-
rapidity (e.g., with time projection chambers) while particle-type-insensitive
detectors are placed at forward and backward rapidities to measure particle
“hits” (e.g., with calorimeters). When measuring phenomena driven by angular
momentum within the QGP (e.g. global $\vec{S}_{\rm H}$ alignment with
$\hat{J}$), QGP byproducts are reconstructed at mid-rapidity while
${\vec{J}_{\mathrm{spec}}}$ is measured using the azimuthal distribution of
forward-/backward-going particles as an approximation of
${\vec{J}_{\mathrm{part}}}$; however, such an approximation is subject not
only to the effects seen in Fig. 3 but also to the experimental constraints of
incomplete detector coverage and imperfect detector efficiencies. Because of
this limitation, random fluctuations will play a larger role and we might
expect the correlation between ${\hat{J}_{\mathrm{spec}}}$ and
${\hat{J}_{\mathrm{mid}}}$ to be smaller than the correlation between
${\hat{J}_{\mathrm{spec}}}$ and ${\hat{J}_{\mathrm{part}}}$. This effect is
shown in Fig. 4 within the AMPT model. For mid-central collisions,
${\hat{J}_{\mathrm{mid}}}$ is well aligned with ${\hat{J}_{\mathrm{syst}}}$
when considering $|y|<2$; however, the degree of alignment drops substantially
when considering the region $|y|<1$ typically used in experimental studies.
This is striking; if taken at face value, this would translate to a correction
of roughly 25% on the observable of interest.

At larger collision energies, the fraction of emitted particles that lie in
the rapidity window $|y|<1$ becomes smaller; we might therefore expect the
correlator
$\left<{\hat{J}_{\mathrm{mid}}}\cdot{\hat{J}_{\mathrm{spec}}}\right>$ to
become smaller with increasing $\sqrt{s_{\mathrm{NN}}}$. On the other hand,
the fraction of emitted particles that lie in the spectator (forward) rapidity
region becomes larger and will impact ${\hat{J}_{\mathrm{spec}}}$. In Fig. 5
we see that, despite this,
$\left<{\hat{J}_{\mathrm{mid}}}\cdot{\hat{J}_{\mathrm{spec}}}\right>$ becomes
smaller as $\sqrt{s_{\mathrm{NN}}}$ becomes larger. This correlation depends
strongly on $\sqrt{s_{\mathrm{NN}}}$, differing by more than a factor of 2
between the lowest and highest collision energies. In Fig. 6,
$\left<{\hat{J}_{\mathrm{mid}}}\cdot{\hat{J}_{\mathrm{spec}}}\right>$ is shown
for mid-central collisions, defined in a number of ways that yield very
similar results, as a function of $\sqrt{s_{\mathrm{NN}}}$. Any
$\sqrt{s_{\mathrm{NN}}}$-dependent experimental observable driven by angular
momentum within the QGP would be corrected in such a manner, by
$\left<{\hat{J}_{\mathrm{mid}}}\cdot{\hat{J}_{\mathrm{spec}}}\right>^{-1}$.
Similarly, it is important for model predictions to use
${\hat{J}_{\mathrm{part}}}$ or ${\hat{J}_{\mathrm{mid}}}$, rather than
${\hat{J}_{\mathrm{syst}}}$, when calculating phenomena driven by angular
momentum within the QGP.

The global spin polarization of hyperons, $\overline{P}_{\mathrm{H}}$, is one
such observable that would require correction, and experimental measurements
of $\overline{P}_{\mathrm{H}}$ Abelev  _et al._ (2007); Adamczyk  _et al._
(2017); Adam  _et al._ (2018) are shown alongside the correlator
$\left<{\hat{J}_{\mathrm{mid}}}\cdot{\hat{J}_{\mathrm{spec}}}\right>$ in Fig.
6. Both $\overline{P}_{\mathrm{H}}$ and
$\left<{\hat{J}_{\mathrm{mid}}}\cdot{\hat{J}_{\mathrm{spec}}}\right>$ fall
with increasing $\sqrt{s_{\mathrm{NN}}}$, though $\overline{P}_{\mathrm{H}}$
demonstrates a stronger dependence. The ratio of $\overline{P}_{\mathrm{H}}$
to $\left<{\hat{J}_{\mathrm{mid}}}\cdot{\hat{J}_{\mathrm{spec}}}\right>$ is
shown in the bottom panel of Fig. 6; the scaled $\overline{P}_{\mathrm{H}}$
demonstrates a notably weaker dependence on $\sqrt{s_{\mathrm{NN}}}$. In
practice, a more detailed study, specific to a detector’s coverage and
acceptance or to a model’s assumptions, would need to be performed to apply a
correction. Without performing such corrections, there will be an apparent
dependence on $\sqrt{s_{\mathrm{NN}}}$ driven at least in part by the behavior
observed in Fig. 6.

Figure 4: The correlation between ${\hat{J}_{\mathrm{mid}}}$ and
${\hat{J}_{\mathrm{spec}}}$ becomes smaller as we further constrain the size
of the rapidity window used for the calculation of ${\hat{J}_{\mathrm{mid}}}$
where initial-state fluctuations play a larger role. Experiments typically are
limited to $|y|<1$. Recall that $|y|=2$ is our participant-spectator cutoff in
AMPT, so that $\vec{J}_{\mathrm{mid},|y|<2}={\vec{J}_{\mathrm{part}}}$. Figure
5: The correlation between ${\hat{J}_{\mathrm{mid}}}$ and
${\hat{J}_{\mathrm{spec}}}$ becomes smaller as we increase
$\sqrt{s_{\mathrm{NN}}}$, where a given rapidity window includes a smaller
fraction of emitted particles and initial-state fluctuations play a larger
role. This is similar to the effects driving the observation in Fig. 4. The
values of $\sqrt{s_{\mathrm{NN}}}$ are chosen to match those of the RHIC Beam
Energy Scan (BES). Figure 6: Panel (a) shows that the correlation between
${\hat{J}_{\mathrm{mid}}}$ and ${\hat{J}_{\mathrm{spec}}}$ for mid-central
collisions (the event class used when studying angular-momentum-driven
phenomena) falls with $\sqrt{s_{\mathrm{NN}}}$. $3<b<8$ fm describes the
region from Fig. 5 where the correlation is flat, and $6.5<b<10.3$ fm and
$7.5<b<8.5$ fm are two ways of approximating 20-50% central collisions.
Experimental results of global hyperon polarization Abelev  _et al._ (2007);
Adamczyk  _et al._ (2017); Adam  _et al._ (2018), $\overline{P}_{\mathrm{H}}$,
are shown alongside these calculations. Also shown, in panel (b), are these
results scaled by the relevant correlation calculated with AMPT.

##  IV ${\hat{J}_{\mathrm{part}}},{\hat{J}_{\mathrm{mid}}}$Correlations with
geometry

The overlap region of a heavy-ion collision is roughly elliptic on average,
with the major axis of the ellipse somewhat aligned with $\hat{y}$. We can fit
the participant coordinates to an ellipse in order to determine its
orientation and study the correlation between ${\hat{J}_{\mathrm{part}}}$ and
this orientation. We characterize the initial shape through harmonic-
eccentricity coefficients $\varepsilon_{n}$ and event-plane angles $\Phi_{n}$
Qiu and Heinz (2012):

$\varepsilon_{n}{e}^{in\Phi_{n}}=-\frac{\int
rdrd\phi{r}^{n}{e}^{in\phi}e(r,\phi)}{\int rdrd\phi{r}^{n}e(r,\phi)}.$ (3)

By taking $n=2$ and treating the initial energy-density distribution
$e(r,\phi)$ as a sum of $\delta$ functions, each at the position of a nucleon,
this reduces to

$\Phi_{2}=\frac{1}{2}\left[\arctan\frac{\sum_{i}{r_{\perp,i}}^{2}\sin(2\phi_{i})}{\sum_{i}{r_{\perp,i}}^{2}\cos(2\phi_{i})}+\pi\right],$
(4)

where $r_{\perp}$ and $\phi$ are the polar coordinates of the participant
nucleons in the transverse plane, as measured from the center of mass of the
participants. This procedure is only applicable in the MCG model where all
nucleons are either considered to be participants or spectators. In the AMPT
model, we can reconstruct the orientation of the elliptic overlap region
responsible for the mid-rapidity region by taking advantage of “elliptic
flow”; the pressure gradient is larger along the shorter axis of the ellipse
than it is along the longer axis. Because of this, the azimuthal distribution
of emitted particles in a rapidity window will reveal the orientation of the
relevant overlap region Voloshin  _et al._ (2010):

$\Phi_{2}=\frac{1}{2}\mathrm{atan2}\left(\sum_{i}w_{i}\sin(2\phi_{i}),\sum_{i}w_{i}\cos(2\phi_{i})\right),$
(5)

where the weight, $w_{i}$, is typically the transverse momentum,
$p_{\mathrm{T}}$.

For central collisions, the overlap region is quite circular and for very
peripheral collisions only a small number of nucleons participate; in both
cases, initial-state fluctuations play a large role in the orientation of the
elliptic fit and therefore on $\Phi_{2}$. In mid-central collisions, the
overlap region is sufficiently elliptic and there are enough participants that
initial-state fluctuations will be sub-dominant; we might therefore expect
$\Phi_{2}$ to be best aligned with $\phi_{\hat{J}_{\mathrm{syst}}}\pm\pi/2$
for mid-central collisions. Such behavior is apparent in the solid lines in
Fig. 7.

We might also intuitively make the naïve assumption that the somewhat elliptic
participant region would be spinning about its major axis and therefore expect
better alignment between $\Phi_{2}$ and
$\phi_{\hat{J}_{\mathrm{part}}},\phi_{\hat{J}_{\mathrm{mid}}}$ than between
$\Phi_{2}$ and $\phi_{\hat{J}_{\mathrm{syst}}}$. If this were true, then the
problematic suppression of
${\hat{J}_{\mathrm{mid}}}\cdot{\hat{J}_{\mathrm{spec}}}$ discussed in Sec. III
could potentially be avoided by measuring ${\hat{J}_{\mathrm{mid}}}$ directly
from $\Phi_{2}$; however, when considering again Fig. 7, there is apparently a
significant suppression of the correlation between $\Phi_{2}$ and
$\phi_{\hat{J}_{\mathrm{part}}},\phi_{\hat{J}_{\mathrm{mid}}}$. This can be
understood by dividing a given tilted elliptic overlap region in two,
lengthwise, and considering that one half is dominated by positive-rapidity
nucleons while the other is dominated by negative-rapidity nucleons. By
applying the right-hand rule to these two halves it is clear that
${\hat{J}_{\mathrm{part}}}$ will tilt to the left as the elliptic overlap
region tilts to the right, and vice versa. One can see in Fig. 1 a few
examples of the major axis tilting away from ${\hat{J}_{\mathrm{part}}}$ over
a range of $b$.

Figure 7: The correlation between the orientation of the elliptic overlap
region and ${\hat{J}_{\mathrm{syst}}}$ is largest for mid-central collisions,
in line with expectations. Counter-intuitively, however, there is a suppressed
correlation between the orientation of the ellipse and
${\hat{J}_{\mathrm{part}}}$. The absolute value of
$\sin(\Phi_{2}-\phi_{\hat{J}_{\mathrm{part}}})$ is used since $\Phi_{2}$ is
physically indistinguishable from $\Phi_{2}\pm\pi$. Recall that $|y|=2$ is our
participant-spectator cutoff in AMPT, shown as the lower two lines, so that
$\vec{J}_{\mathrm{mid},|y|<2}={\vec{J}_{\mathrm{part}}}$. The horizontal
dashed line represents $2/\pi$, which is the average absolute value of the
sine of the difference between two random, uncorrelated numbers.

For the same reasons that we expected
${\hat{J}_{\mathrm{mid}}}\cdot{\hat{J}_{\mathrm{spec}}}$ to decrease both when
reducing the size of the window in $y$ and when increasing
$\sqrt{s_{\mathrm{NN}}}$, we might expect the correlation
$\left<\left|\sin(\Phi_{2}-\phi_{\hat{J}_{\mathrm{mid}}})\right|\right>$ to
again decrease in AMPT when considering $|y|<1$, as well as when considering
larger $\sqrt{s_{\mathrm{NN}}}$. While not shown in this paper, we indeed did
find such additional suppressions to this correlation. This decorrelation
between $\Phi_{2}$ and
$\phi_{\hat{J}_{\mathrm{part}}},\phi_{\hat{J}_{\mathrm{mid}}}$ demonstrates
that one can not avoid the corrective factors discussed in Sec. III by
indirectly measuring ${\hat{J}_{\mathrm{mid}}}$ through $\Phi_{2}$.

##  V Summary

Initial-state fluctuations drive a decorrelation between
${\hat{J}_{\mathrm{syst}}}$ and
${\hat{J}_{\mathrm{part}}},{\hat{J}_{\mathrm{mid}}}$, which is the largest for
central and peripheral collisions. Conservation of angular momentum further
suppresses this correlation between ${\hat{J}_{\mathrm{spec}}}$ and
${\hat{J}_{\mathrm{part}}},{\hat{J}_{\mathrm{mid}}}$, albeit slightly. Only
${\hat{J}_{\mathrm{spec}}}$ is experimentally accessible, and is used as an
approximation of ${\hat{J}_{\mathrm{mid}}}$. As the size of the mid-rapidity
window becomes smaller, the correlation between ${\hat{J}_{\mathrm{spec}}}$
and ${\hat{J}_{\mathrm{mid}}}$ is suppressed further. Similarly, this
decorrelation becomes more dramatic with increasing $\sqrt{s_{\mathrm{NN}}}$.

The orientation of the elliptic overlap region, $\Phi_{2}$, has a smaller
correlation with ${\hat{J}_{\mathrm{part}}},{\hat{J}_{\mathrm{mid}}}$ than
with ${\hat{J}_{\mathrm{syst}}}$, in conflict with potentially intuitive
expectations. The correlation between the orientation of elliptic shape and
${\hat{J}_{\mathrm{part}}}$ is further suppressed when constraining the mid-
rapidity window to $|y|<1$, as well as when increasing
$\sqrt{s_{\mathrm{NN}}}$. Deducing ${\hat{J}_{\mathrm{part}}}$ from $\Phi_{2}$
is therefore not a viable method to avoid the problems of correlation
suppression between ${\hat{J}_{\mathrm{mid}}}$ and
${\hat{J}_{\mathrm{spec}}}$.

The findings presented here hold significant implications for measurements of
phenomena driven by angular momentum within the QGP, and particularly for
those interested in the dependence on $\sqrt{s_{\mathrm{NN}}}$. Based on our
model-dependent study, it is crucial for studies of angular-momentum-driven
phenomena, such as experimental measurements of $\overline{P}_{\mathrm{H}}$,
to correct for the decorrelation between ${\hat{J}_{\mathrm{spec}}}$ and
${\hat{J}_{\mathrm{mid}}}$ in a $\sqrt{s_{\mathrm{NN}}}$-dependent manner, and
for model predictions of such phenomena to use ${\hat{J}_{\mathrm{mid}}}$
instead of ${\hat{J}_{\mathrm{syst}}}$. Without these corrections, any
observed dependence will be driven at least in part by this decorrelation.

##  VI Acknowledgements

We thank Jinfeng Liao and Giorgio Torrieri for helpful conversations. This
work was supported by the U.S. Department of Energy grant DE-SC0020651.

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# Rainbow Stars and Rota’s Basis Conjecture for Graphic Matroids

(Date: October 2023)

###### Abstract.

Let $G$ be a connected multigraph with $n$ vertices, and suppose $G$ has been
edge-colored with $n-1$ colors so that each color class induces a spanning
tree. Rota’s Basis Conjecture for graphic matroids posits that one can find
$n-1$ mutually edge-disjoint rainbow spanning trees. In a recent paper,
Maezawa and Yazawa have shown that the conjecture holds if one assumes that
the color classes induce spanning stars. We delve further into the star case
to explore some extreme subcases including: all stars with different centers,
the same center, or one of two centers. In addition, we identify the cases in
which a graph composed of monochromatic stars can be decomposed into rainbow
stars. We also show that the statement is false if one replaces ‘stars’ with
‘paths’.

Anant Asthana, Shreev Goyal

Mentored by Dr. Anton Dochtermann

##  1\. Introduction

Rota’s basis conjecture is a well-known open question in matroid theory that
involves rearranging elements in a given set of bases to produce other
‘rainbow’ bases. The statement of the conjecture first appeared in a paper of
Huang and Rota [1]. For the case of graphic matroids (which is our main focus
here), the conjecture can be stated as follows.

######  Conjecture 1.1.

Let $G$ be a connected graph on $n$ vertices, and suppose $G$ has been edge
colored with $n-1$ colors so that each color class induces a monochromatic
spanning tree. Then $G$ can be decomposed into $n-1$ disjoint rainbow spanning
trees.

We refer to Section 2 for any undefined terms. Note that Rota’s Basis
Conjecture is vacuous if the underlying graph $G$ is simple (the relevant
number of disjoint spanning trees cannot exist), so here we are always working
with graphs that have parallel edges.

According to Davies and McDiarmid [2], graphs with no $K_{4}$ minor satisfy
the conjecture. In a recent preprint, Maezawa and Yazawa [3] established
Rota’s Basis Conjecture for graphic matroids in the special case that each
color class induces a monochromatic _spanning star_. They prove that under
this assumption $G$ can be decomposed into $n-1$ disjoint rainbow trees. Even
for this special case, their proof is rather long and technical.

In this note we study special cases of Conjecture 1.1 involving arrangements
of monochromatic stars and how one can find resulting rainbow spanning stars
or trees. In particular, we consider the case of all stars having different
centers, and the case of all stars having their centers among at most two
vertices.

Our main result states that if the color classes induce monochromatic stars,
then one can find a collection of disjoint rainbow _stars_ if and only if all
the given stars share the same center or all have different centers. See
Theorem 3.6.1. We also consider the case of induced paths and show, via a
counterexample, that a similar statement does not hold in this case. See
Proposition 4.1 for details.

Along the way, we also consider the question of _how many_ collections of
rainbow trees (or stars) can be found in a given edge colored graph (Rota’s
Conjecture states that at least one exists). We see that in some cases only
one collection of rainbow stars can be found (see Proposition 3.1), and in
other cases we observe a connection to Latin squares (see Theorem 3.2.3).

##  2\. Preliminaries

For us, a graph $G$ consists of a vertex set $V$ and a multiset of edges
$E\subset V\times V$ where $e_{vw}\in E$ implies $w\neq v$ and $e_{wv}\in E$.
Hence, our graphs are undirected and have no loops, but may (and usually will)
have multiple edges. Unless otherwise specified, our graphs will be connected.

A _tree_ in $G$ is a set of edges $\tau\subset E$ with the property that the
subgraph induced by $\tau$ has no closed paths (so in particular $\tau$ has no
multiple edges). A _star_ in $G$ is a tree $S\subset E$ with the property that
the subgraph induced by $S$ has a single vertex (called an _center_) that is
incident to every edge in $S$. A tree $\tau$ of $G$ is said to be _spanning_
(in $G$) if every vertex $v\in V$ is incident to some edge of $\tau$. Note
that if $G$ is connected and has $n$ vertices, then a tree $\tau$ is spanning
if and only if it consists of $n-1$ edges.

We will be interested in _edge-colorings_ of a graph $G$, by which we mean a
function $f:E\rightarrow[m]$ where $m$ is some positive integer and
$[m]=\\{1,2,\dots,m\\}$. A subgraph $H\subset G$ is said to be _monochromatic_
if every edge is assigned the same value, and is _rainbow_ if all edges are
assigned different values.

Rota’s Basis Conjecture tells us that if we are given $n-1$ monochromatic
spanning trees on $n$ vertices, we can always find a set of $n-1$ rainbow
spanning trees. For instance, in the example below, a graph on $n=4$ vertices
composed of the top 3 monochromatic trees can be decomposed into the bottom 3
rainbow trees. Thus, this graph satisfies Rota’s Basis Conjecture.

In fact, Rota’s Basis Conjecture can be seen as the ’Trees-to-Trees’ extension
of ’Stars-to-Stars’ that we make precise below.

######  Definition 2.1.

Suppose $G$ is a graph on $n$ vertices that has been edge colored with $n-1$
colors so that each monochromatic subgraph induces a star. Then we say Stars-
to-stars holds for $G$ if one can find a collection of $n-1$ disjoint rainbow
stars.

##  3\. Stars-to-Stars and Stars-to-Trees

###  3.1. All Stars Have a Different Center

######  Theorem 3.1.1.

Stars-to-Stars holds if all monochromatic stars have a different center.

###### Proof.

Suppose $G$ is a connected graph on vertex set $V=\\{v_{1},\ldots,v_{n}\\}$
and edge set $E$. Suppose $G$ has been edge colored with $n-1$ colors so that
each color class induces a monochromatic star. Without loss of generality,
suppose $v_{1},\dots,v_{n-1}$ are the internal nodes of these stars.

Now, we claim that this graph can be decomposed into $n-1$ rainbow stars. Let
$c_{j}$ denote the color of the monochromatic star with center at $v_{j}$; we
will denote the whole monochromatic star as $S_{j}$. To construct
$\textsc{RS}_{j}$, the rainbow star with center at vertex $j$, we do the
following:

  1. (1)

We choose the edge $e_{jn}\in S_{j}$ to connect node $v_{n}$ to $v_{j}$ such
that $e_{jn}$ has color $c_{j}$.

  2. (2)

Then, for any node $v_{i}$ such that $i\neq n,j$, we choose the edge
$e_{ji}\in S_{i}$ between $v_{i}$ and $v_{j}$ to connect node $i$ to $j$.

Thus,

$\textsc{RS}_{j}=\\{V\cup e_{ji}\mid i\neq n,i\neq j,e_{ji}\in S_{i},\text{
and }e_{jn}\in S_{j}\\}.$

For instance, consider the following graph on $n=4$ vertices:

The nodes of this graph have already been rearranged as detailed above. The
center vertex has also been colored black to distinguish it from the centers
of the monochromatic spanning stars. Notice that this graph is induced by the
following $3$ monochromatic stars:

In addition, the center of each star has been colored with the same color as
the edges of that monochromatic star. We follow the algorithm outlined in the
proof to dissect the construction into 3 rainbow stars, as shown below.

Note that all of these rainbow stars are disjoint and use all the edges of the
original graph. In other words, all the rainbow stars are built by connecting
a colored point and the center point with its own color, and then receiving a
color from every other colored point. We can guarantee that all edges are used
exactly once. Thus, the proof is complete. ∎

### Example with 5 nodes

The same process can be used to decompose a graph with $5$ vertices (induced
by $4$ monochromatic spanning stars) to create $4$ rainbow stars, as depicted
below.

Shown above are the $4$ monochromatic stars that comprise the graph in the
previous figure.

Following the algorithm described in the previous proof, we connect each
center of a $c$ color-induced star to the center vertex, and we connect that
same center to the remaining vertices using the respective colors of the stars
whose center is at that remaining vertex.

This algorithm can be used for an arbitrary number of points.

In the context of Rota’s Conjecture, a natural question to ask is _how many_
sets of rainbow spanning objects one can find in certain contexts. In the case
of finding rainbow stars from monochromatic stars, we have the following
observation.

######  Proposition 3.1.

Suppose $G$ is a graph on $n$ vertices that has been edge colored with $n-1$
colors so that the induced monochromatic subgraphs are all stars with
different centers. Then there exists a unique collection of $n-1$ disjoint
spanning rainbow stars.

###### Proof.

If we take a closer look, we can identify a few points to consider:

  1. (a)

The center node can not have a star because if it did, this would use up the
edges connecting the center and the remaining trees could not connect the
center.

  2. (b)

Each of the colored stars on their nodes contain $n-1$ edges of its own color
and 1 color edge from the other $n-2$ stars.;

  3. (c)

Each star is created by combining the lone edge to the center plus the
collection of edges from the other stars.

This means there is one and exactly one way to create each star, so the total
number of collections of disjoint spanning rainbow stars is one. ∎

###  3.2. All Stars Have the Same Center

######  Theorem 3.2.1.

Stars-to-Stars holds if all monochromatic stars have the same center.

###### Proof.

Say a graph $G=(V,E)$ satisfies the conditions of the statement. We will
construct the resulting rainbow stars. Let $v_{i}$ denote any given vertex of
the graph, with $v_{0}$ signifying the common center of all the monochromatic
stars. For $1\leq j\leq n-1$, we define the $j$th resulting star
$RS_{j}\subset E$ in the following manner:

$RS_{j}=\\{e_{v_{0}v_{j+k-1}}\mid e_{v_{0}v_{j+k-1}}\in S_{k},1\leq k\leq
n-1\\},$

where $j+k-1$ is taken mod $n-1$. ∎

For instance, in the example below on 4 vertices and 3 colors, the following
trees result. Notice how the colors are merely rotated between trees.

Once again we are interested in how many sets of rainbow spanning trees can be
found in this context, and in particular we consider the following.

######  Question 3.2.2.

From a configuration with all monochromatic stars sharing the same center, how
many collections of $n-1$ spanning rainbow stars are possible?

Returning to the $n=4$ example where there are 3 rainbow stars, there exist
exactly 2 distinct unordered collections, as illustrated below.

For $n=2,3$, the number of collections is $1$. For $n=4$, the number of
collections increases to $2$. For $n=5$, the number of collections jumps to
24. $n=6$ yields an even larger result that is much more difficult to
calculate by hand (see table).

In fact, we claim the following:

######  Theorem 3.2.3.

For a graph on $n$ vertices where all induced spanning monochromatic stars
share the same center, the number of collections of $n-1$ disjoint spanning
rainbow stars is

$\Omega(n)=\frac{L_{n-1}}{(n-1)!},$

where $L_{n-1}$ is the number of Latin squares of size $n-1$.

###### Proof.

The strategy involves representing the set of all edges of all stars as a
matrix.

Consider each resulting star as a row of $n-1$ elements, where the $i$th
element in the row is the color of the edge connecting $v_{0}$ and $v_{i}$.
Considering all the resulting rainbow stars at once, we can combine $n-1$
rows/stars of $n-1$ elements each into a rainbow star matrix. In other words,
the $i$th star would be $i$th row in the matrix, and the $j$th edge in the
$i$th star would be the $j$th element in the $i$th row.

For instance, in the example shown below, the leftmost star would represent
the row $\\{R,B,G\\}$, the next would be $\\{G,R,B\\}$, and the last would be
$\\{B,G,R\\}$, where R=red, B=blue, and G=green.

This gives us the matrix:

$\begin{bmatrix}R&B&G\\\ G&R&B\\\ B&G&R\end{bmatrix}$

Notice that this rainbow star matrix possesses two special properties:

  1. (1)

Each element within a given row is distinct.

  2. (2)

Each element within a given column is distinct.

The first condition is due to the fact that every edge within a rainbow star
must have a different color by definition. The second condition holds because
an edge of a given color connecting the same pair of vertices cannot be used
in two different rainbow stars (as per the definition of decomposition).

Thus, such a rainbow star matrix is really just a Latin square of size $n-1$
in disguise. Therefore, the number of such rainbow star matrices is $L_{n-1}$,
i.e. the number of size $n-1$ Latin squares. However, we ignore the order of
the resulting trees, as this would be rearranging stars in a collection which
wouldn’t count as a distinct collection. If we were to fix a color onto the
diagonal of the rainbow star matrix, (in the above case red), then the real
number of collections would be ${L_{n-1}}$ divided by the number of ways to
permute the number of rows, $(n-1)!$ Thus, the number of ways to decompose a
graph consisting of $n-1$ monochromatic stars sharing a center into rainbow
stars is $\frac{L_{n-1}}{(n-1)!}$. ∎

An explicit formula due to Shao and Wei [4] for ${L_{n}}$ is given below.

$L_{n}=n!\sum_{A\epsilon{B_{n}}}(-1)^{\sigma_{0}(A)}\binom{{\text{per
}A}}{{n}}$ (1)

Here, $B_{n}$ is the set of $n\times n$ matrices with entries in $\\{0,1\\}$,
$\sigma_{0}(A)$ is the number of zero elements in $A$, and $\text{per }A$ is
the permanent of the matrix $A$. Fitting this into our equation
$\frac{L_{n-1}}{(n-1)!}$, the $(n-1)!$s cancel out nicely, leaving us with:

$\Omega(n)=\sum_{A\epsilon{B_{n-1}}}(-1)^{\sigma_{0}(A)}\binom{{perA}}{{n-1}}$
(2)

Values for $\Omega(n)$ grow very quickly, as shown below.

n | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8
---|---|---|---|---|---|---|---|---
$\Omega(n)$ | 1 | 1 | 1 | 2 | 24 | 1344 | 1128960 | 12198297600

###  3.3. Extension of 3.2 to Trees

######  Theorem 3.3.1.

Suppose $G$ has been edge colored so that each color class induces $n-1$
copies of the same monochromatic spanning tree. Then Rota’s Basis Conjecture
holds.

Our argument stated in section 3.1 isn’t restricted to just stars with the
same center case. It can be extended to any collection of monochromatic
spanning trees that are all the same. This is because for each edge on a
rainbow tree, we pick edges from a set of $n-1$ edges on $n-1$ trees.
Therefore, the same conversion to a rainbow matrix holds, and the Latin Square
argument for identical stars can be used for identical trees (identical except
for color, of course). The only difference is that we have to assign each edge
to a column in the matrix, giving us a rainbow tree matrix.

We present an example of how our argument applies to a set of identical trees
when $n=4$.

The two resulting collections of rainbow trees would look like this.

For the matrix, we let the first column be the left edge, the middle column to
be the diagonal edge, and the right column to be the right edge. The
corresponding rainbow tree matrices would be the following, where the left
matrix represents the top collection and the right matrix represents the
bottom matrix.

$\begin{bmatrix}R&B&G\\\ G&R&B\\\ B&G&R\end{bmatrix}\&\begin{bmatrix}R&G&B\\\
B&R&G\\\ G&B&R\end{bmatrix}$

This method of counting the number of collections will hold true as long as
all monochromatic spanning trees within a collection are identical.

###  3.4. Stars-to-stars Can Fail

We next consider a situation where monochromatic stars do not lead to rainbow
stars.

Consider a graph $G$ on $4$ vertices, which is edge-colored in such a way that
two of the induced monochromatic stars share a center. Below, we see that the
red and blue stars share the top left corner as their center. We also run into
the problem where we can’t color our vertex according to its star, so we’ll
just mix the colors of the stars with internal vertices at that spot,
conveniently red and blue make purple. We will mark the bottom left corner
with a black vertex.

There is only one possible way to decompose $G$ into 3 rainbow trees, ignoring
symmetry between the red and blue edges:

Notice that only one of the trees above is a rainbow star. Thus, it is
impossible to decompose $G$ into rainbow stars.

###  3.5. All Stars Have a Center that Falls on One of Two Nodes

Although an arbitrary arrangement of monochromatic stars does not necessarily
lead to a collection of rainbow stars, we can give another proof of Rota’s
Conjecture in a special case.

######  Theorem 3.5.1.

Suppose a graph $G$ on $n$ vertices is edge colored such that it consists of
$n-1$ monochromatic spanning stars that have a center on one of two vertices.
Then, it is possible to decompose $G$ into $n-1$ disjoint spanning rainbow
trees.

###### Proof.

Let’s suppose we have a graph $G$ with $n_{1}$ monochromatic stars centered on
vertex $1$ and $n_{2}$ monochromatic stars centered on vertex $2$, where
$n_{1}+n_{2}=n-1$. We want to decompose such a graph into $n-1$ rainbow trees.
We will construct these $n-1$ rainbow trees from scratch.

First, some notation: let $C_{1},C_{2}$ denote the set of colors that belong
to $1$ and $2$, respectively. Then, $\left|C_{1}\right|=n_{1}$ and
$\left|C_{2}\right|=n_{2}$.

For the vertices that serve as the centers of stars, we define an arbitrary
’order’ of colors. For instance, ’for vertex $k$, blue comes clockwise after
red’. We will also reorder the colors within $C_{k}$ and $C_{j}$ to reflect
this arbitrary choice of color ordering. Then, construct $n-1$ trees using the
following idea:

  * •

For each tree, we choose the edge $e_{jk}$, where $c$ is distinct for each
tree.

  * •

Pick any one of the trees-under-construction. Say that for one given tree, the
edge connecting $k$ and $j$ is of the form $e_{jk}$, where $e_{jk}\in S_{j}$.
Then, for the $n_{1}-1$ remaining colors in $C_{j}$, we place edges of the
form $e_{j\delta}$, where $\delta=k+i$ and $e_{j\delta}\in S_{i}$; this way,
the order of the colors of the edges emanating from $j$ matches the order that
was arbitrarily determined previously.

  * •

After that, we will be left with $(n-1)-(n_{1})=n_{2}$ points that are still
not connected to a vertex. To rescue those points, we will connect them each
to $k$; the order of the colors of the edges will follow the pre-determined
order.

Thus, we see a ’rotation’ motif present throughout such constructs.

For instance, say we have a graph on $n=5$ vertices that consists of the
following monochromatic stars.

5431254312

5431254312Clockwise: $C_{1}=Red>Blue,\;\;C_{2}=Green>Yellow$

Then, we can start by laying down the $k-j$ connections.

5431254312

5431254312

If the edge connecting $k$ to $j$ is color $c$ and $c$ belongs to $k$, we
connect points to $k$ with a specific color depending on the pre-determined
color ordering with respect to $c$.

5431254312

5431254312

Finally, we follow a similar process for $j$ but with the remaining vertices
and follow the clockwise pre-determined ordering.

5431254312

5431254312

To see why this idea does not produce any cycles or disconnections, note the
following points:

  * •

Each vertex $m\neq j,k$ is connected to exactly one of $j$ or $k$ (either $m$
was connected to $j$ through the second step, or it was one of the remaining
$n_{2}$ vertices that was connected to $k$ through the third step). Also,
every tree, by construction, has an edge connecting $j$ to $k$. Therefore,
there are no disconnected points in our constructed rainbow trees.

  * •

Because there are no repeated edges or edges of the form $e_{mr}$ in the
resulting rainbow trees, where $m,r\neq k,j$, there are no cycles.

We can also guarantee that the collection of rainbow trees is disjoint; if one
were given the original set of monochromatic trees and the orderings of the
stars within a center, then it would be possible to tell which rainbow tree an
edge would belong to. This is possible because for a center, we can place the
remaining edges of the other colors emanating from that center based off of
the ordering of stars on that center, and then place the remaining edges of
the stars with the other center based off of the ordering of stars on that
center. Being able to dictate which tree an edge is in guarantees
disjointedness.

Thus, we can decompose $G$ into $n-1$ disjoint spanning rainbow trees. ∎

###  3.6. When Stars-to-Stars Holds

Sections 3.1 and 3.2 suggest that it is possible to make rainbow stars from
monochromatic stars with different or same centers, respectively, but a
question to ponder is if these are the only cases of monochromatic stars which
produce rainbow stars. A natural extension to the star problem addressed above
is that of whether a graph induced by monochromatic spanning paths can be
decomposed into a set of rainbow spanning paths.

######  Theorem 3.6.1.

Stars-to-stars holds if and only if all monochromatic stars share the same
center or all have different centers.

###### Proof.

The ‘if’ implication has been established in Theorem 3.1.1 and Theorem 3.2.1.
We are left to prove the converse.

The statement is trivial for $n=2$ because there is only $1$ star consisting
of $1$ edge. For $n\geq 3$, say we start out with a decomposition of a graph
into monochromatic stars. In said decomposition, say there are $s_{k}$ stars
on vertex $k$. Then the degree of vertex $k$ is

$D_{k}=s_{k}(n-1)+(n-1)-s_{k}=(s_{k}+1)(n-2)+1.$

The $s_{k}(n-1)$ comes from the number of stars on that center times the
number of edges per star, and the $(n-1)-s_{k}$ comes from the number of stars
not on that center times $1$ for the number of edges that connect to vertex
$k$ per star. The right-hand side is a factored version of the equation.

Now, consider a hypothetical decomposition of the same graph into rainbow
stars. There must be exactly $s_{k}$ rainbow stars centered at vertex $k$. If
there were more than $s_{k}$ rainbow stars on vertex $k$, we would have at
least $(s_{k}+2)(n-2)+1$ edges incident to vertex $k$, which is larger than
$D_{k}$. Likewise, if there were less than $s_{k}$ rainbow stars on vertex
$k$, we would have at most $(s_{k})(n-2)+1$ edges incident to vertex $k$,
which is smaller than $D_{k}$.

Now, note that if a monochromatic star is centered at $k$, then all edges of
that color emanate from vertex $k$. If that monochromatic star is not centered
at $k$, then only one edge of that color connects to vertex $k$. Since each
rainbow star must have exactly one of a certain color, exactly $0$, $1$, or
$n-1$ rainbow stars must be centered at vertex $k$. There being $0$ stars on a
vertex must be included because there are $n$ vertices and $n-1$ stars, and by
the Pigeonhole Principle, at least one vertex does not have a star centered at
it. Therefore, $s_{k}=0,1,n-1$, proving our theorem. Note, the $1$ or $n-1$
rainbow stars being centered at a point corresponds with Theorems 3.1.1 and
3.2.1, respectively. ∎

##  4\. Paths-to-Paths

######  Definition 4.1.

A spanning path on a set of $n$ vertices is a set of $n-1$ edges such that

  1. (a)

All vertices are included in the path (hence the name “spanning”), and

  2. (b)

For a given spanning path, all vertices have degree at most 2.

In other words, a spanning path can be drawn in one motion without lifting the
pencil off of the paper.

######  Definition 4.2.

Suppose $G$ is a graph on $n$ vertices that has been edge colored with $n-1$
colors so that each monochromatic subgraph induces a spanning path. Then we
say Paths-to-Paths holds for $G$ if one can find a collection of $n-1$
disjoint spanning rainbow paths.

We next show that Paths-to-Paths is in general not possible.

######  Proposition 4.1.

For the edge coloring of the graph $G$ on $4$ vertices depicted below, there
does not exist any collection of $3$ disjoint rainbow paths.

###### Proof.

Suppose $G$ is the edge colored graph on $4$ nodes whose color classes induce
the 3 monochromatic spanning paths shown below.

Notice that the first vertex has only 3 total edges emanating from it (1 red,
1 blue, and 1 green). Thus, each of the 3 rainbow spanning paths must have the
first vertex be either a starting or ending vertex; without loss of
generality, we assume the first vertex to be the starting node for all 3 of
the rainbow spanning paths.

Now, consider constructing the rainbow spanning path that uses the red edge
between the first node and the third node.

The remaining edges in the rainbow spanning path must be either blue or green.
Thus, the two possibilities for the rainbow spanning path are as presented
below.

However, notice that neither of the two options presented above is a path;
both of them are stars. Therefore, the original graph cannot be decomposed
into 3 rainbow spanning paths, establishing the claim. ∎

##  5\. Further thoughts

We end with some discussion and ideas for future research. Further work
regarding Conjecture 1.1 would include considering the following questions.

  * •

If the colors classes induce monochromatic spanning paths, under which
circumstances can we find spanning rainbow _paths_ (i.e. when does Paths-to-
Paths hold)?

  * •

Can we prove Conjecture 1.1 for graphs where the color classes induce
monochromatic spanning _paths_ (i.e. does Paths-to-Trees hold)?

We also remark that Rota’s Basis Conjecture makes sense in the more general
context of matroid theory. Recall that a _matroid_ is a pair $(M,E)$
consisting of a finite ground set $E$ and a set $M$ of subsets of $E$, called
_independent sets_ , satisfying:

  1. (a)

$\emptyset\in M$;

  2. (b)

If $X\in M$ and $Y\subset X$ then $Y\in M$;

  3. (c)

If $X,Y\in M$ and $|X|>|Y|$, then there exists an element $x\in X\setminus Y$
so that $Y\cup\\{x\\}\in B$.

The maximal elements of $M$ are called the _bases_ of $M$. The _rank_ of $M$
is the cardinality of any (and hence all) basis element. Rota’s Conjecture can
then be stated in full generality as follows.

######  Conjecture 5.1.

Suppose $M$ is a rank $n$ matroid with a collection of disjoint bases
${\mathcal{B}}=\\{B_{1},\dots,B_{n}\\}$. Then one can arrange the elements of
${\mathcal{B}}$ into an $n\times n$ matrix in such a way that the $i$th row
consists of the elements of $B_{i}$ and each column is also a basis of $M$.

In other words, if we think of each $B_{i}$ as a color class, we obtain a
collection of _colorful_ bases ${\mathcal{C}}=\\{C_{1},\dots,C_{n}\\}$ where
each $C_{i}$ contains exactly one element from each $B_{j}\in{\mathcal{B}}$.
In this paper, we considered matroids coming from a graph $G=(V,E)$, where the
ground set is given by the set $E$ of edges, and independent sets are given by
collections of edges that do contain a cycle.

Conjecture 5.1 has been established for all paving matroids by Geelen and
Humphries in [5], and for matroids of rank at most $3$ by Chan [6]. It was
shown by Wild [7] that a stronger version of the conjecture holds for
_strongly base orderable_ matroids. This class is closed under duality and
taking minors, and includes all gammoids.

In [8] it was shown that for any matroid one can always find
$(1/2-\text{o}(1))n$ disjoint transversal bases (so that we are ‘halfway’ to
Rota’s Conjecture). Recently, Rota’s Basis Conjecture for realizable matrices
over finite fields has been established in a probabilistic setting by
Sauermann [9]. Interest in Rota’s Basis Conjecture continues to this day, as
evidenced by the recent Polymath project [10]. It would be interesting to
consider Rota’s Basis conjecture for other classes of matroids.

## Acknowledgements

We would like to thank the 2022 Mathworks Honors Summer Math Camp for their
support and encouragement. We would like to extend our thanks to Dr. Max
Warshauer and Dr. Eugene Curtin of Texas State University for connecting us
after said camp. We would finally like to thank Dr. Anton Dochtermann of Texas
State University for continually supporting and guiding us throughout the
research project and providing a great working environment.

## References

  * [1] Huang, R. and Rota G.-C., On the relations of various conjectures on Latin squares and straightening coefficients, Discrete Math. 128 (1994), no. 1-3, pp. 225–236.
  * [2] Davies, J. and McDiarmid, C. (1976), Disjoint Common Transversals and Exchange Structures. Journal of the London Mathematical Society, s2-14: pp. 55–62.
  * [3] Maezawa, Shun-ichi; Yazawa, Akiko; Special Case of Rota’s Basis Conjecture on Graphic Matroids. Electron. J. Combin. 29 (2022), no. 3, Paper No. 3.
  * [4] Shao, Jia-yu and Wei, Wan-di. A Formula for the Number of Latin Squares. Discrete Mathematics, 110 (1992), pp. 293–296.
  * [5] Geelen, J and Humphries, P. J. (2006), Rota’s basis conjecture for paving matroids, SIAM Journal on Discrete Mathematics, 20 (2006), pp. 1042–1045.
  * [6] Chan, Wendy, _An exchange property of matroid_ , Discrete Mathematics, 146 (1995), pp. 299–302.
  * [7] Wild, M. On Rota’s problem about n bases in a rank n matroid, Adv. Math. 108 (1994), no. 2, pp. 336-–345.
  * [8] Bucć, Matija; Kwan, Matthew; Pokrovskiy, Alexey; Sudakov, Benny Halfway to Rota’s basis conjecture. Int. Math. Res. Not. (2020), no. 21, pp. 8007–8026.
  * [9] Sauermann, Lisa. Rota’s basis conjecture holds for random bases of vector spaces. arXiv preprint arXiv:2203.17121 (2022).
  * [10] tchow8. Rota’s Basis Conjecture: Polymath 12, post 3. https://polymathprojects.org/2017/05/05/rotas-basis-conjecture-polymath-12-post-3/ (2017).

A. Asthana, Dulles High School, Sugar Land, Texas

E-mail address<EMAIL_ADDRESS>
S. Goyal, McNeil High School, Austin, Texas

E-mail address<EMAIL_ADDRESS>
A. Dochtermann, Department of Mathematics, Texas State University, San Marcos,
Texas

E-mail address<EMAIL_ADDRESS>