reactions can take place into bounded domains Vazquez (2010). We note,
incidentally, that Smoluchowski’s rate constant (9) cannot describe even one-
sink reaction inside a larger spherical domain since it does not consider
diffusive interaction effect due to the boundary of englobed domain.

In Ref. Ovchinnikov, Timashev, and Belyy (1989) authors concluded that their
system of equations (3.97) solves the $N$-sink diffusion problem under
Smoluchowski boundary conditions exactly. However, results obtained in Sec.
VIII show that the system (3.97) has been derived for the monopole
approximation and, therefore, the above conclusion is incorrect.

It is interesting to note that, clustering of sinks was found to reduce the
absorbing rate even more significantly compared with nonclustered systems
Richards (1986). The method gives us possibility to tackle trapping reactions
in several clusters of obstacles and sinks in the same way as for separated
sinks.

Taking into account our previous results Traytak (1992, 2013) and Proposition
X.1 we can conclude that diffusive interaction "glues" fully absorbing sinks
effectively onto a fully absorbing effective sink. Therefore the whole
$N$-sink array behaves like a single, isolated fully absorbing sink of some
characteristic size.

For a number of applied tasks, one should use the derived microscopic solution
(65) (or (66)) along with microscopic trapping rates (67) calculating
appropriate configurationally averaged values to obtain effective macroscopic
values associated with heterogeneous media comprising different types of sinks
Brailsford (1976); Loewenberg and Gavalas (1988); Borodin (1994); Torquato
(2002).

Finally, let us dwell on one more possible application of the approach at
issue. In Ref. Traytak (2020) we have established for the first time that even
classical passive phoresis of microparticles (i.e., motion due to a constant
gradient of some scalar field) should be treated by means of their interaction
with so-called surrounding bodies, which are responcible for the external
gradient. In this context, it is worth to note that the first and only attempt
to apply a similar approach for studing particle interactions for
thermophoresis was undertaken by Keh and Chen in Ref. Keh and Chen (1995).
They used so-called polyadic tensors but in fact, these tensors are the same
as the ICT up to a constant factors. Again the GMSV in terms of ICT can be
used to investigate these quite a challenging problems.

##  XII Conclusions and Perspectives

Motivated through new applications in physics, chemistry and biology this
paper deals with the generalized method of separation of variables by means of
the irreducible Cartesian tensors formalism extending our previous study on
the diffusion-controlled reactions to the general case of diffusion-influenced
reactions. We have developed here a complete modification of the above
mentioned method which essentially hinges on a number useful properties of the
irreducible Cartesian tensors. Formally we reduced the steady-stae diffusion
problem to a set of $N$ Laplace equations with respect to partial solution
together with the prescribed $N$ boundary conditions.

In order to determine the place and significance of our research we have
presented a rather comprehensive review of literature devoted to the
theoretical methods for diffusion-influenced reactions on spherical sinks.
Besides, we paid close attention to the mathematical aspects of the
appropriate diffusion problem, namely: (a) rigorous statement of the problem,
including terminology clarification, (b) refining solution algorithm and (c)
justification of the method convergence. The formulation of the geometric part
of the diffusion boundary value problems was performed the most fully than
anywhere else. We highlighted the importance of a concept of a smooth manifold
for the future theoretical investigations on diffusion-influenced processes
occuring in domains with multiply connected boundaries. We have given a proof
of the irregular to regular translation addition theorem in terms of the
irreducible Cartesian tensors which seems to be the simplest among previously
known theorem for solid harmonics in terms of polar spherical coordinates. For
the first time corresponding solution algorithm was divided into eight clear
steps. Then we implemented this algorithm step by step in full detail to solve
the posed diffusion problem. A convergence analysis was conducted to verify
the possibility of simple-iteration method usage in order to solve appropriate
resolving infinite system of linear algebraic equations. It turns out that the
method of reduction may be used to solve numerically the resolving system to
any degree of accuracy even when the simple-iteration method is inapplicable.
Wherein the desired accuracy of the solution can be achieved just by
increasing the truncation degree. Furthermore, note that the method has
advantages such as shorter calculation time and more accurate results compared
to purely numerical methods.

Our results fully confirm known Hinsen-Felderhof statement that it is more
preferred to work with harmonic functions using their irreducible Cartesian
tensor form throughout diffusion-reaction problems even for ensembles of
spherical sinks.

This research provides also a better understanding of the reactants diffusion
among arrays comprising arbitrary collection of arbitrary many fully
absorbing, reflecting and partially reflecting sinks from both mathematical
and physical viewpoints.

The approach developed here is applicable to a wide class of reaction-
diffusion systems, providing useful tests for numerical calculations along
with approximate analytical estimates of the local concentration field and
reaction rates in many specific cases of practical importance.

Finally it is important to highlight that numerical methods can handle
diffusion problems for complex geometries in unbounded domains, yet they
require to introduce an articial boundary, reducing the original exterior
domain to a bounded domain. The generalized method of separation of variables
does not have this drawback and provides simple direct numerical and
analytical calculations. So we can obtain approximations as the truncated
point multipole expansions up to the monopole, dipole, quadrupole etc. order.

It is essential to note that self-phoresis of the Janus particles moving due
to a chemical reaction in the surrounding fluid is investigated by means of
monopole approximation Stark (2018). Using the approach considered here this
important problem may be attack taking into account multipole corrections
Traytak (2020).

The last but not least: in the present study we clearly show that the theory
of 3D differentiable manifolds is an appropriate mathematical tool to describe
diffusion and reactions in domains with multiply connected boundaries. In this
context, we note that, rigorously posing and solving the appropriate diffusion
problems, we, nevertheless, endeavored to avoid considering a number of
inherent subtle mathematical points. This issues will be a topic for the
future work.

## Appendix. Background on the irreducible Cartesian tensors

To make this paper maximally self-contained and facilitate applications in
diffusion-influenced reactions theory, we provide here necessary mathematical
notations, definitions and facts, required in rigorous formulation and
applications of the ICT technique.

###  XII.1 Definitions and examples

It is well-known that there exits two alternative approaches to harmonic
functions theory based on either Laplace’s representation of solid harmonics
with respect to the polar spherical coordinates or Maxwell’s theory of poles
given in terms of the successive derivatives of fundamental solution written
for a Cartesian coordinate system Courant and Hilbert (1953); Snider (1998).
The latter approach may be formulated with the aid of the irreducible
Cartesian tensors.

Generally speaking, the irreducible tensors transform according to irreducible
representation of the special rotation group $SO\left(3\right)$. In other
words the irreducible tensors like spherical harmonics form a basis in the
representation space of group $SO\left(3\right)$ (see XII.3). To our
knowledge, the ICT simple mathematical description suitable for applied
physicists has been provided in Ref. Coope, Snider, and McCourt (1965).
Moreover, for many years there has been a significant number of publications
devoting to the use of the ICT formalizm in a wide variety of applications
(see, e.g., Refs. Brenner (1964); Anderson (1974); Cipriani and Silvi (1982);
Martynov (1998) and references therein). As regards adopted notations and
definitions, in this paper we basically follow known books by Hess Hess (2015)
and Snider Snider (2018), where comprehensive discussions on the various
facets of irreducible Cartesian tensors are presented.

A real tensor field of $n$-th order defined on ${\mathbb{R}}^{3}$ we denote by
$\bm{T}_{n}\left(\mathbf{r}\right)$ in coordinate-free notation. The same
tensor field can be also described by its components with respect to a fixed
orthonormal basis $\\{{\mathbf{e}}_{\alpha}\\}_{\alpha=1}^{3}$ given in
$\\{O;\mathbf{r}\\}$, i.e.
${\bm{T}}_{n}\left({\mathbf{e}}_{\gamma_{1}},\ldots,{\mathbf{e}}_{\gamma_{n}}\right)=T_{{\gamma_{1}}\ldots{\gamma_{n}}}\left(\mathbf{r}\right)$.
Here and in the following,
$T_{{\gamma_{1}}\ldots{\gamma_{n}}}\left(\mathbf{r}\right)$ are $3^{n}$
components of $n$-rank Cartesian tensor field
$\bm{T}_{n}\left(\mathbf{r}\right)$. It is clear that the action of the nabla
operator with $\partial_{r_{\mu}}:=\nabla_{\mu}$ to a field, which is a tensor
of rank $l$, yields a tensor of rank $l+1$. We recall in passing that one
should not distinguish between co- and contravariant index due to duality of
Cartesian basis Snider (2018).

The product symbol $\odot^{q}$ represents a full tensorial contraction of
multiplicity $q$ or shortly the $q$-fold contraction between two tensors with
the convention ${\bm{S}}_{0}\odot{\bm{T}}_{0}:={\bm{T}}_{0}$. Particularly,
for tensors ${\bm{S}}_{n}$ and ${\bm{T}}_{m}$ and $q\leq n,m$ we have a tensor

${\bm{S}}_{n}\odot^{q}{\bm{T}}_{m}:=\sum_{\gamma_{1}}\ldots\sum_{\gamma_{q}}S_{\gamma_{1}\ldots\gamma_{n}}\odot^{q}T_{\mu_{1}\ldots\mu_{m}}\,$

of rank $n+m-2q$, whereas it is a scalar if $n=m=q$.

Any tensor, which cannot be reduced to a tensor of lower rank is known as an
irreducible tensor Hess (2015). While a general (reducible) $n$-th order
Cartesian tensor ${\bm{T}}_{n}$ has $3^{n}$ components, corresponding
irreducible tensor besides the rank $n$ is likewise characterized by its
weight $j$ Coope, Snider, and McCourt (1965). It is common knowledge that
latter tensor has $2j+1$ independent components, which form the basis of the
$j$ weight irreducible representation of the full 3D special rotation group
$SO\left(3\right)$ including mirror Snider (2018).

Furthermore, among all ICT obtained from a given $n$ rank tensor by means of
reductions there exists one unique irreducible tensor with the same weight and
rank $n=j$. It is known in literature as ICT in natural form Coope, Snider,
and McCourt (1965). We deal with irreducible tensors in natural form because
they best suited to apply within the scope of the GMSV.

Thus, for $n$ rank Cartesian tensors our focus lies on their irreducible parts
in natural form at that we use notations
$\mathop{{\bm{T}}_{n}}\limits^{\rule[-2.8903pt]{0.72229pt}{3.61371pt}\rule{10.84006pt}{0.72229pt}\rule[-2.8903pt]{0.72229pt}{3.61371pt}}$
and
$\mathop{T_{{\gamma_{1}}\ldots{\gamma_{n}}}}\limits^{\rule[-2.8903pt]{0.72229pt}{3.61371pt}\rule{24.57153pt}{0.72229pt}\rule[-2.8903pt]{0.72229pt}{3.61371pt}}$,
where symbol
$\mathop{\left(\>...\>\right)}\limits^{\rule[-2.8903pt]{0.72229pt}{3.61371pt}\rule{18.06749pt}{0.72229pt}\rule[-2.8903pt]{0.72229pt}{3.61371pt}}$
stands for the irreducible part of a Cartesian tensor ${\bm{T}}_{n}$ of rank
$n\geq 2$ with respect to the special rotation group $SO(3)$ Hess (2015);
Snider (2018).

The irreducible tensor in natural form is appeared to be fully symmetric and
traceless, therefore, this allows us to use rather simple definition of the
irreducible Cartesian tensors Junk, Budday, and Bohlke (2012); Mane (2016).

######  Definition XII.1.

A Cartesian tensor of rank $n\geq 2$ is called irreducible Cartesian tensor,
if it is: (i) totally symmetric under an arbitrary permutation of the indices;
and (ii) traceless under the contraction of any pair of indices.

An important point is that according to definition (XII.1) the ICT can be
expressed in terms of successive partial derivatives of the generating
function $1/r$ for all points in
$\mathbb{R}^{3}\backslash\left\\{{\mathbf{0}}\right\\}$.

######  Proposition XII.1.

An irreducible Cartesian tensors of rank $n$ (when $n\geq 2$) may be defined
as Courant and Hilbert (1953); Hess (2015)

$\mathop{\hskip 5.69046pt{r}_{\gamma_{1}}{\ldots
r}_{\gamma_{n}}}\limits^{\rule[-2.8903pt]{0.72229pt}{3.61371pt}\rule{32.52127pt}{0.72229pt}\rule[-2.8903pt]{0.72229pt}{3.61371pt}}=\frac{\left(-1\right)^{n}}{\left(2n-1\right)!!}r^{2n+1}\partial_{r_{\gamma_{1}}}\ldots\partial_{r_{\gamma_{n}}}\left(\frac{1}{r}\right)\,.$
(A.1)

Here $r_{\gamma_{\nu}}$ are Cartesian coordinates of a vector $\mathbf{r}$
($\gamma_{\nu}=\overline{1,3}$), and $\left(2n-1\right)!!:=1\cdot
3\cdot\ldots\cdot\left(2n-1\right)$ indicates the double factorial for odd
natural numbers with the convention $(-1)!!:=1$.

###### Proof.

Check that the tensors introduced by Eq. (A.1) are symmetric for any pair of
indexes and traceless.

(i) Indeed, Eq. (A.1) involves directly that the introduced tensor is
symmetric with respect to permutation of its indexes

$\mathop{\hskip
5.69046pt{r}_{\gamma_{1}}\ldots{r}_{\alpha}\ldots{r}_{\beta}\ldots{r}_{\gamma_{n}}}\limits^{\rule[-2.8903pt]{0.72229pt}{3.61371pt}\rule{83.11005pt}{0.72229pt}\rule[-2.8903pt]{0.72229pt}{3.61371pt}}=\mathop{\hskip
5.69046pt{r}_{\gamma_{1}}\ldots{r}_{\beta}\ldots{r}_{\alpha}\ldots{r}_{\gamma_{n}}}\limits^{\rule[-2.8903pt]{0.72229pt}{3.61371pt}\rule{83.11005pt}{0.72229pt}\rule[-2.8903pt]{0.72229pt}{3.61371pt}}\,.$
(A.2)

(ii) Moreover, due to harmonicity of the generating function

${\bm{\nabla}}^{2}\left(\frac{1}{r}\right)=\delta_{\alpha{\beta}}\odot^{2}\partial_{r_{\alpha}}\partial_{r_{\beta}}\left(\frac{1}{r}\right)=0\quad\mbox{for}\quad
r>0\,$ (A.3)

tensors (A.1) are also traceless with respect to any pair of their indexes

$\delta_{\alpha{\beta}}\odot^{2}\mathop{\hskip
5.69046pt{r}_{\gamma_{1}}\ldots{r}_{\alpha}\ldots{r}_{\beta}\ldots{r}_{\gamma_{n}}}\limits^{\rule[-2.8903pt]{0.72229pt}{3.61371pt}\rule{83.11005pt}{0.72229pt}\rule[-2.8903pt]{0.72229pt}{3.61371pt}}=0\,,$
(A.4)

where $\delta_{\alpha{\beta}}$ is the Kronecker delta. Thus Eq. (A.1)
satisfies both points of definition (XII.1). ∎

######  Remark XII.1.

Note furthermore that traces cannot be defined for any scalars
$a_{0}\in{\mathbb{R}}$ (tensors of rank $0$) or vectors
$\left({r}_{1},{r}_{2},{r}_{3}\right)$ (tensors of rank $1$)). Nethertheless
they are harmonic and, therefore, irreducible by definition Hess (2015).

In applications are also used the multipole potentials defined by:
$\bm{X}_{n}^{\mp}\left(\mathbf{r}\right)={X}^{\mp}_{\gamma_{1}\ldots\gamma_{n}}\left({\bf
r}\right)$ Hess (2015), where components read

$\displaystyle{X}^{-}_{\gamma_{1}\ldots\gamma_{n}}\left({\bf
r}\right):={\left(-1\right)^{n}}\partial_{r_{\gamma_{1}}}\ldots\partial_{r_{\gamma_{n}}}\left(\frac{1}{r}\right)\,,$
(A.5) $\displaystyle{X}^{+}_{\gamma_{1}\ldots\gamma_{n}}\left({\bf
r}\right):=\left(2n-1\right)!!\mathop{\hskip 5.69046pt{r}_{\gamma_{1}}{\ldots
r}_{\gamma_{n}}}\limits^{\rule[-2.8903pt]{0.72229pt}{3.61371pt}\rule{32.52127pt}{0.72229pt}\rule[-2.8903pt]{0.72229pt}{3.61371pt}}\,.$
(A.6)

Owing to definition they are connected by the relation

$\displaystyle\bm{X}_{n}^{+}\left(\mathbf{r}\right)=r^{(2n+1)}\bm{X}_{n}^{-}\left(\mathbf{r}\right)\,.$
(A.7)

For reasons explained below (see XII.3), it is quite natural to call
potentials $\bm{X}_{n}^{-}$ and $\bm{X}_{n}^{+}$ irregular and regular ICT,
respectively Hinsen and Felderhof (1992). However, throughout this paper we
use ICT defined by Eq. (A.1), since they are most extensively employed in
applications.

Denote by $d^{2}{\hat{r}}$ the magnitude of the surface element on the unit
sphere in global coordinates

$\displaystyle\partial\Omega_{1}\equiv\partial\Omega\left({\bm{0}};1\right):=\\{{\mathbf{r}}\in\mathbb{R}^{3}:{\hat{r}}=1\\}\,.$
(A.8)

Then for all $n\geq 0$ we can write the following orthogonality condition Hess
(2015)

$\displaystyle\oint_{\partial\Omega_{1}}\mathop{{\hat{r}}_{\gamma_{1}}\ldots{\hat{r}}_{\gamma_{n}}}\limits^{\rule[-2.8903pt]{0.72229pt}{3.61371pt}\rule{33.96698pt}{0.72229pt}\rule[-2.8903pt]{0.72229pt}{3.61371pt}}\cdot\mathop{{\hat{r}}_{\mu_{1}}\ldots{\hat{r}}_{\mu_{k}}}\limits^{\rule[-2.8903pt]{0.72229pt}{3.61371pt}\rule{35.41269pt}{0.72229pt}\rule[-2.8903pt]{0.72229pt}{3.61371pt}}d^{2}{\hat{r}}=\frac{4\pi
n!}{\left(2n+1\right)!!}\delta_{nk}\Delta^{(n)}_{\gamma_{1}\ldots\gamma_{n},\mu_{1}\ldots\mu_{n}}\,,$
(A.9)

where in case of sphere $\hat{\mathbf{n}}=\hat{\mathbf{r}}$ (in component
notation $\hat{r}_{\alpha}=r_{\alpha}/r$) is a unit vector identifying a point
on $\partial\Omega_{1}$ and
$\left.\mathop{{r}_{\gamma_{1}}\ldots{{r}}_{\gamma_{n}}}\limits^{\rule[-2.8903pt]{0.72229pt}{3.61371pt}\rule{34.68929pt}{0.72229pt}\rule[-2.8903pt]{0.72229pt}{3.61371pt}}\right|_{\partial\Omega_{1}}=\mathop{{\hat{r}}_{\gamma_{1}}\ldots{\hat{r}}_{\gamma_{n}}}\limits^{\rule[-2.8903pt]{0.72229pt}{3.61371pt}\rule{33.96698pt}{0.72229pt}\rule[-2.8903pt]{0.72229pt}{3.61371pt}}$
is the restriction of the ICT to the unit sphere (A.8). In Eq. (A.9)
$\Delta^{(n)}_{\gamma_{1}\ldots\gamma_{n},\mu_{1}\ldots\mu_{k}}$ stands for
the $2n$ rank projector, which project any tensor of rank $n$ into its
irreducible part. Particularly, three first projectors are Hess (2015)

$\displaystyle\Delta^{(0)}=1\,,\quad\Delta^{(1)}=\delta_{\alpha{\beta}}\,,$
$\displaystyle\Delta^{(2)}_{\alpha\beta,\alpha^{\prime}\beta^{\prime}}=\frac{1}{2}\left(\delta_{\alpha\alpha^{\prime}}\delta_{\beta\beta^{\prime}}+\delta_{\alpha\beta^{\prime}}\delta_{\beta\alpha^{\prime}}\right)-\frac{1}{3}\delta_{\alpha\beta}\delta_{\alpha^{\prime}\beta^{\prime}}\,.$

###  XII.2 Efimov’s recurrent formula

In XI.1 we have already mentioned that an important advantage of the ICT over
solid harmonics is relatively simple recurcion formulas obtained by rather
easy successive differentiation of generating function. Interested readers are
encouraged to refer to works, where explicit recurrent expressions for the ICT
are comprehensively presented Cipriani and Silvi (1982); Blaive and Metzger
(1984); Applequist (1989); Hinsen and Felderhof (1992); Ehrentraut and Muschik
(1998); Hess (2015); Snider (2018). Here let us dwell briefly on a little-
known, but simple and powerful approach proposed by Efimov Efimov (1979, 2022)
that allowed us to calculate ICT.

To derive the relevant Efimov’s recurrent formula first we introduce an
auxiliary tensor field

$\displaystyle\Gamma_{\gamma_{1}...\gamma_{n}}\left({\bf
r}\right):={\left(-1\right)^{n}}{X}^{+}_{\gamma_{1}\ldots\gamma_{n}}\left({\bf
r}\right)\,.$ (A.10)

Inversion of the radius vector $\mathbf{r}\rightarrow\mathbf{r}r^{-2}$ leads
to an isomorphic mapping of the gradient operator in the inverted space of
harmonic functions (with opposite sign), i.e. the vector operator
$\widehat{\mathbf{D}}=-r^{-1}{\nabla_{1/r}}r$. Direct calculations yield the
expression

$\widehat{\mathbf{D}}:=2\mathbf{r}\left(\mathbf{r}\cdot\nabla\right)-r^{2}\nabla+\mathbf{r}\,.$
(A.11)

With the help of this operator it may be shown that the following recursion
holds

$\Gamma_{0}=1,\quad\Gamma_{\gamma_{n}\gamma_{1}...\gamma_{n-1}}^{\left(n\right)}=\widehat{D}_{\gamma_{n}}\cdot\Gamma_{\gamma_{1}...\gamma_{n-1}}^{\left(n-1\right)}\quad\mbox{for}\quad
n\in\mathbb{N}.$

Taking this relations into account one easily derives the desired Efimov’s
formula Efimov (1979)

$\displaystyle\Gamma_{\gamma_{1}...\gamma_{n}}=\widehat{D}_{\gamma_{1}}\widehat{D}_{\gamma_{2}}...\widehat{D}_{\gamma_{n}}\cdot
1\,.$ (A.12)

Hence, in the explicit form the first several irreducible Cartesian tensors of
ranks $n$ corresponding to the monopole ($n=0$), dipole ($n=1$), quadrupole
($n=2$) and octupole ($n=3$) potentials read as follows:

$\displaystyle\mathop{\hskip
5.69046pta_{0}}\limits^{\rule[-2.8903pt]{0.72229pt}{3.61371pt}\rule{9.39545pt}{0.72229pt}\rule[-2.8903pt]{0.72229pt}{3.61371pt}}:=a_{0}\,,\mathop{\hskip
5.69046pt{r}_{\gamma_{1}}}\limits^{\rule[-2.8903pt]{0.72229pt}{3.61371pt}\rule{9.39545pt}{0.72229pt}\rule[-2.8903pt]{0.72229pt}{3.61371pt}}:={r}_{\gamma_{1}}\,,\mathop{\hskip
5.69046pt{r}_{\gamma_{1}}{r}_{\gamma_{2}}}\limits^{\rule[-2.8903pt]{0.72229pt}{3.61371pt}\rule{21.68121pt}{0.72229pt}\rule[-2.8903pt]{0.72229pt}{3.61371pt}}={r}_{\gamma_{1}}{r}_{\gamma_{2}}-\frac{1}{3}r^{2}\delta_{\gamma_{1}{\gamma_{2}}}\,,$
$\displaystyle\mathop{\hskip
5.69046pt{r}_{\gamma_{1}}{r}_{\gamma_{2}}{r}_{\gamma_{3}}}\limits^{\rule[-2.8903pt]{0.72229pt}{3.61371pt}\rule{31.07556pt}{0.72229pt}\rule[-2.8903pt]{0.72229pt}{3.61371pt}}={r}_{\gamma_{1}}{r}_{\gamma_{2}}{r}_{\gamma_{3}}-\frac{1}{5}r^{2}\left({r}_{\gamma_{1}}\delta_{\gamma_{2}{\gamma_{3}}}+{r}_{\gamma_{2}}\delta_{\gamma_{1}{\gamma_{3}}}+{r}_{\gamma_{3}}\delta_{\gamma_{1}{\gamma_{2}}}\right)\,.$

###  XII.3 Connection with solid harmonics

As indicated in Sec. II, solid harmonics were thoroughly used previously to
investigate many-sink effects during diffusion-influenced reactions. So, it is
expedient to point to an intimate relationship between them and the ICT Stone
(1975); Normand and Raynal (1982); Dennis (2004); Hess (2015).

Recall that real solid harmonics $\psi_{nm}\left(r,\theta,\phi\right)$, where
the integer $m$ such that $-n\leq m\leq n$ and $n=\overline{0,\infty}$, are
harmonic functions in spherical coordinates $\\{O;r,\theta,\phi\\}$ Tikhonov
and Samarskii (1963). For brevity we shall denote them as
$\psi_{n}\left(r,\theta,\phi\right)$. Further, according to their behavior at
the origin $O$, two kinds of solid harmonics are distinguished:

(1) irregular at $O$ solid harmonics
${\psi}_{n}^{-}\left(r,\theta,\phi\right)$ has a pole of order $n+1$
singularity at $O$;

(2) regular at $O$ solid harmonics ${\psi}_{n}^{+}\left(r,\theta,\phi\right)$
are real-valued harmonic homogeneous polynomials of order $n$.

Hence, irregular solid harmonics ${\psi}_{n}^{-}$ automatically satisfies
regularity condition at infinity (27) and, therefore, they are regular at
infinity.

Clearly, irregular $\bm{X}_{n}^{-}$ (A.5) and regular $\bm{X}_{n}^{+}$ (A.6)
ICT correspond to irregular and regular solid harmonics ${\psi}_{n}^{-}$ and
${\psi}_{n}^{+}$, respectively.

Represent tensors $\bm{X}_{n}^{\pm}\left({\bf r}\right)$ in the form most
closely resemble classical irregular
${\psi}_{n}^{-}\left(r,\theta,\phi\right)$, regular
${\psi}_{n}^{+}\left(r,\theta,\phi\right)$ solid and real spherical
${Y}_{n}\left(\theta,\phi\right)$ harmonics:

$\displaystyle{X}^{-}_{\gamma_{1}\ldots\gamma_{n}}\left({\bf
r}\right)=r^{-\left(n+1\right)}Y_{{\gamma_{1}}\ldots{\gamma_{n}}}\left({\hat{\mathbf{r}}}\right)\,,$
(A.13) $\displaystyle{X}^{+}_{\gamma_{1}\ldots\gamma_{n}}\left({\bf
r}\right)=r^{n}Y_{{\gamma_{1}}\ldots{\gamma_{n}}}\left({\hat{\mathbf{r}}}\right)\,,$
(A.14) $\displaystyle
Y_{{\gamma_{1}}\ldots{\gamma_{n}}}\left({\hat{\mathbf{r}}}\right):=\left(2n-1\right)!!\mathop{{\hat{r}}_{\gamma_{1}}\ldots{\hat{r}}_{\gamma_{n}}}\limits^{\rule[-2.8903pt]{0.72229pt}{3.61371pt}\rule{32.52127pt}{0.72229pt}\rule[-2.8903pt]{0.72229pt}{3.61371pt}}\,.$
(A.15)

It is apparent that the ICT
$Y_{{\gamma_{1}}\ldots{\gamma_{n}}}\left({\hat{\mathbf{r}}}\right)$ like
spherical harmonics depend on the direction of vector $\mathbf{r}$ only Hess
(2015).

As noted above, the ICT similar to solid harmonics form a basis in the
representation space of the group $SO\left(3\right)$, therefore, there exists
a non-singular linear transformation and its inverse in space
$\mathbb{R}^{2n+1}$ which map these basises into each other Courant and
Hilbert (1953). Using known Sylvester’s theorem Courant and Hilbert (1953);
Dennis (2004) one can find the desired connection between tensors
$\bm{X}_{n}^{\pm}$ and classical real solid harmonics ${\psi}_{n}^{\pm}$
explicitly. We emphasize here that original Sylvester’s proof is rather
sophisticated Courant and Hilbert (1953), therefore, interested readers are
encouraged to refer to a simple proof of Sylvester’s theorem proposed by
Backus in Ref. Backus (1970).

Thus, it can be shown that Sylvester’s theorem implies

$\displaystyle\frac{1}{r^{n+1}}Y_{n}\left(\theta,\phi\right)=\sum_{m=0}^{n}\sum_{l=0}^{m}c_{lm}\partial_{r_{1}}^{l}\partial_{r_{2}}^{m-l}\partial_{r_{3}}^{n-m}\left(\frac{1}{r}\right)\,,$
(A.16) $\displaystyle\text{where}\hskip
6.00006ptY_{n}\left(\theta,\phi\right)=\sum_{m=0}^{n}Y_{n}^{m}\left(\theta,\phi\right)\,,$
$\displaystyle Y_{n}^{m}\left(\theta,\phi\right):=\left(a^{m}_{n}\cos
m\phi+b^{m}_{n}\sin m\phi\right)P^{m}_{n}\left(\cos\theta\right)\,$ (A.17)

is the real spherical harmonic of degree $n$ and order $m$. In Eq. (A.17)
$a^{m}_{n}$, $b^{m}_{n}$ and $c_{lm}$ are some known real coefficients and
$P^{m}_{n}\left(\cos\theta\right)$ are associated Legendre functions James
(1968).

Particularly, for the axially symmetric case ($m=0$), taking all derivatives
along $Or_{3}$ axis, we have
$r_{\gamma_{1}}=r_{\gamma_{2}}=\ldots=r_{\gamma_{n}}=r_{3}$ and formula (A.16)
boils down to well-known relation Courant and Hilbert (1953)

$\displaystyle\frac{1}{r^{n+1}}P_{n}\left(\cos\theta\right)=\frac{\left(-1\right)^{n}}{n!}{\partial^{n}_{r_{3}}\left(\frac{1}{r}\right)}\,.$
(A.18)

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# Distinguishing regular graphs

Jakub Kwaśny  AGH University of Science and Technology,
Faculty of Applied Mathematics,
al. Mickiewicza 30, 30-059 Krakow, Poland  Marcin Stawiski111 Corresponding
author<EMAIL_ADDRESS>AGH University of Science and Technology,
Faculty of Applied Mathematics,
al. Mickiewicza 30, 30-059 Krakow, Poland

###### Abstract

Call an edge colouring of a graph _distinguishing_ if the only automorphism
that preserves that colouring is the identity. We prove that every locally
finite connected regular graph of order at least 6 admits a distinguishing
edge colouring with two colours.

##  1 Introduction

All colourings in this paper are edge colourings unless stated otherwise. An
edge colouring $c$ of graph $G$ _preserves_ an automorphism $\varphi$ of $G$
if each edge is mapped by $\varphi$ to an edge of the same colour. If $c$ does
not preserve $\varphi$, then we say that $c$ _breaks_ $\varphi$. If $c$ breaks
all non-trivial automorphisms of $G$, then we say that $c$ is
_distinguishing_. The _distinguishing index_ of $G$ is the least number of
colours in a distinguishing edge colouring of $G$. We denote it by
$D^{\prime}(G)$. Note that distinguishing index is well-defined for every
connected graph except $K_{2}$. The analogous parameter for vertex colourings
is called the _distinguishing number_ of $G$, and it is denoted by $D(G)$.
Distinguishing vertex colourings were introduced by Babai [1] in 1977. They
arose during his study of the graph isomorphism problem, which eventually led
to his proof of the existence of quasi-polynomial algorithm for this problem
(see [2]). Distinguishing colourings in a more general group-theoretic context
were studied first by Cameron, Neumann, and Saxl [3] in 1984. The edge variant
was studied first by Pilśniak and Kalinowski [5]. We note that the problem
whether a given graph has the distinguishing index at most two is closely
related (but not equivalent) to the existence of its asymmetric spanning
subgraph.

The distinguishing index of every finite connected graph $G$ is at most
$\Delta(G)$ unless $G$ is $K_{2}$, $C_{3}$, $C_{4}$, or $C_{5}$. Moreover, the
finite connected graphs for which $D^{\prime}(G)=\Delta(G)$ have been
classified by Pilśniak [9] in 2017. Lehner, Pilśniak and Stawiski [10] proved
in 2020 that the distinguishing index of connected infinite graph $G$ is at
most $\Delta(G)-1$ unless $G$ is a double ray.

Distinguishing index of a graph cannot be much greater than its distinguishing
number as proved by Lehner and Smith [7] in 2020. However, $D^{\prime}(G)$ may
be much smaller than $D(G)$ as in the case of finite complete graphs of order
at least 6 for which $D(G)=|G|$ but $D^{\prime}(G)=2$.

In this paper we are interested in regular graphs, that is, graphs where
$\delta(G)=\Delta(G)$. Lehner, Pilśniak and Stawiski [6] in 2020 proved the
following theorem.

######  Theorem 1.

Let $G$ be a connected $\Delta$-regular graph for some
$\Delta\in\mathbb{N}\setminus\\{1\\}$. Then $D^{\prime}(G)\leq 3$.

Recently, Grech and Kisielewicz [4] generalized this result to graphs for
which minimum degree is at least half the maximum degree.

######  Theorem 2.

Let $G$ be a connected graph with finite maximum degree $\Delta$ and minimum
degree $\delta$. If $\delta>\Delta/2$ and $G\neq K_{2}$, then
$D^{\prime}(G)\leq 2$.

We prove that the following theorem holds for regular locally finite graphs,
which was conjectured in [6].

######  Theorem 3.

If $G$ is a connected locally finite regular graph, then $D^{\prime}(G)\leq
2$, unless $G$ is either $K_{n}$ for $n\leq 5$, or $K_{n,n}$ for $n\leq 3$, or
$C_{5}$.

Pilśniak [9] in 2017 proved that if a finite connected graph $G$ with at least
$7$ vertices contains a Hamiltonian path, then $D^{\prime}(G)=2$. While there
are regular graphs that do not admit Hamiltonian paths, it is not unreasonable
to think that most of them at least have an asymmetric spanning tree. By
Dirac’s Theorem, a finite graph on $n$ vertices whose minimum degree is at
least $\frac{n}{2}$ has a Hamiltonian cycle. Consequently, if $G$ is a finite
regular graph, then either $G$ or its complement has a Hamiltonian cycle, and
one might hope that a Hamiltonian cycle in the complement of $G$ can also be
used to construct an asymmetric spanning tree of $G$. For vertex transitive
connected graphs it is also worth pointing out that Lovász [8] conjectured
that all of them admit a Hamiltonian path. This would imply that
$D^{\prime}(G)\leq 2$ for all such graphs except $K_{2}$ which is a special
case of Theorem 3. Therefore Theorem 3 may be seen as a support for Lovász
Conjecture.

The problem of determining the distinguishing index of regular graphs which
are not locally finite was also previously studied. Lehner, Pilśniak and
Stawiski proved that the distinguishing index of connected $\kappa$-regular
graph is at most $\kappa$ if $\kappa$ is a fixed point of aleph hierarchy.
They also conjectured that this holds for all infinite cardinals $\kappa$.
This was recently proved by Wilson [11] although the paper is not yet
available. Combining Theorem 3 and the theorem proved by Wilson we obtain the
following theorem.

######  Theorem 4.

If $G$ is a connected regular graph of order at least 7, then
$D^{\prime}(G)\leq 2$.

##  2 Proof of the main result

In this section we prove Theorem 3. All colourings in the proof below are edge
colourings.

###### Proof.

Denote $\mathcal{H}=\\{K_{2},K_{3},K_{4},K_{5},K_{3,3},C_{4},C_{5}\\}$. By the
results from [9] and [10], it follows that every connected 2-regular or
3-regular graph $G$ which is not in $\mathcal{H}$ has the distinguishing index
at most 2. Furthermore, it is known that each complete graph of order at least
6 admits a distinguishing edge colouring with two colours. Denote the maximum
degree of $G$ by $\Delta$ (so by the assumptions, every vertex has degree
$\Delta$). We proceed by induction on $\Delta$ with the base case
$\Delta\in\\{2,3\\}$ already discussed. Therefore, we can assume that $G$ is
not complete, that $\Delta\geq 4$, and that any $\Delta^{\prime}$-regular
graph with $\Delta^{\prime}<\Delta$ has a distinguishing edge colouring with
$2$ colours unless it lies in $\mathcal{H}$.

Fix an arbitrary root vertex $r$ in $G$. Automorphisms of the rooted graph
$(G,r)$ are exactly these automorphisms of $G$ which fix $r$. We shall prove
the following claim.

######  Claim 1.

There exists an edge colouring $c$ of $G$ using colours red and blue, with the
following properties:

  1. (i)

$c$ breaks every non-trivial automorphism of $(G,r)$,

  2. (ii)

there exist exactly one or two vertices with all their incident edges coloured
with red; one of them is the root $r$, and the other, if it exists, is a
neighbour of $r$,

  3. (iii)

if there are two such vertices, then for each $k\in\\{1,\ldots,\Delta\\}$ the
vertex $r$ has a neighbour with exactly $k$ incident red edges.

We now show how to construct a distinguishing colouring $c^{\prime}$ of $G$
using the colouring $c$ from the claim.

If $r$ is the only vertex with all incident edges of red colour, then it is
fixed, therefore $c$ is a distinguishing colouring of $G$ and we put
$c^{\prime}=c$. Otherwise, there exists a vertex $x\in N(r)$ which has all its
incident edges coloured with red and we can assume that there exists an
automorphism moving $r$ to $x$. Let $y$ be the unique neighbour of $x$ with
exactly one incident blue edge.

Case 1. The vertex $y$ lies in $N(r)$. We argue that recolouring $yr$ with
blue yields a distinguishing colouring $c^{\prime}$ of $G$. First, notice that
if we recolour any edge incident to $r$ with blue, then the modified colouring
is also distinguishing for $(G,r)$. Hence, after the recolouring, $x$ becomes
fixed as the only vertex with all incident red edges and $r$ is the only
neighbour of $x$ with exactly one incident blue edge.

Case 2. The vertex $y$ is not in $N(r)$. The vertex $y$ has a common blue edge
with exactly one vertex $z$. Let $k\geq 1$ be the number of blue edges
incident to $z$. By the assumption that $\Delta\geq 4$, the vertex $r$ has at
least two neighbours which are distinct from $x$ and $z$. Both of them have
different numbers of incident blue edges, so at least one of this numbers is
different from $k-1$. Define $c^{\prime}$ as obtained from $c$ by recolouring
the edge between the corresponding vertex and $r$ with blue. Again, $x$
becomes the only vertex with all incident red edges, so it is fixed. By the
previous argument, $c^{\prime}$ is distinguishing for $(G,r)$, so we are left
to confirm that $r$ is also fixed. However, $x$ has only two neighbours with
exactly one incident blue edge: $y$ and $r$, while $y$, unlike $r$, has a blue
edge to a vertex with exactly $k$ incident blue edges. Therefore, $x$ cannot
be mapped to $y$ and to any other vertex by an automorphism, so once more we
obtain a distinguishing colouring of $G$ using two colours.

_Proof of Claim 1._

Let $\mathcal{S}_{0}$ be the set of orbits of vertices with respect to
$\operatorname{Aut}(G,r)$. We fix some well-ordering $\leq$ of
$\mathcal{S}_{0}$ such that if $A,B\in\mathcal{S}_{0}$ and vertices of $A$ are
closer to $r$ than $B$ (note that this distance is constant on any element of
$\mathcal{S}_{0}$), then $A\leq B$. In the proof we consider sets of orbits
$\mathcal{S}$ with respect to nested subgroups of $\operatorname{Aut}(G,r)$.
For each such set of orbits we consider a well-ordering $\leq_{\mathcal{S}}$
such that all these orderings are compatible in the following sense: if
$\leq_{\mathcal{S}_{1}}$, $\leq_{\mathcal{S}_{2}}$ are the orderings of the
sets of orbits related to
$\Gamma_{1}\subseteq\Gamma_{2}\subseteq\operatorname{Aut}(G,r)$, respectively,
then for any $A_{1},B_{1}\in\mathcal{S}_{1}$, $A_{1}\subseteq
A_{2}\in\mathcal{S}_{2}$, $B_{1}\subseteq B_{2}\in\mathcal{S}_{2}$,
$A_{2}\leq_{\mathcal{S}_{2}}B_{2}$ we have $A_{1}\leq_{\mathcal{S}_{1}}B_{1}$.
We shall drop the subscript in the ordering sign $\leq_{\mathcal{S}}$ whenever
the related set $\mathcal{S}$ is clear.

Now, let $\mathcal{S}$ be one of such sets of orbits. If we consider some
$S\in\mathcal{S}$, then we refer to an edge between $S$ and vertex from $A$ as
_back edge_ if $A<S$, as _forward edge_ if $S<A$, or as _horizontal edge_ if
$S=A$. Vertices connected to $S\in\mathcal{S}$ by forward or back edges will
be referred to as _forward neighbours_ or _back neighbours_ of $S$,
respectively. Note that by definition of $\mathcal{S}$, every vertex in
$S\in\mathcal{S}$ is incident to the same number $f$ of forward edges, the
same number $b$ of back edges, and the same number $h$ of horizontal edges
(and clearly $f+b+h=\Delta$). Further note that by definition of
$\mathcal{S}_{0}$ and the properties of $\leq$ from the previous paragraph,
$\bigcup\\{S^{\prime}:S^{\prime}<S\\}$ contains all vertices that are closer
to $r$ than $S$ (and perhaps some vertices at the same distance to $r$). Thus,
if $S$ is not a singleton of $r$, then every vertex in $S$ has at least one
back neighbour; i.e. $b\geq 1$.

We now proceed to the construction of the colouring $c$ which breaks all non-
trivial automorphisms of $(G,r)$. First, we colour each edge incident to $r$
with red. Put $\mathcal{S}=\mathcal{S}_{0}$, and denote by $S_{1}$ the least
element of $\mathcal{S}_{0}\setminus\\{\\{r\\}\\}$. Since $S_{1}$ may be the
only of the considered orbits which has $\Delta$ elements, it needs to be
considered separately in some parts of the proof.

We colour the edges of $G$ by iteratively running the procedure defined in the
following paragraphs until all the vertices of $G$ are fixed with respect to
the group $\operatorname{Aut}(G,r)$. The procedure takes the first non-trivial
orbit $S$ (with regard to the well-ordering $\leq$) and colours all the edges
incident to the vertices of $S$, which are not yet coloured, leaving $S$ fixed
pointwise. The procedure needs some assumptions which we shall guarantee to
remain satisfied after its completion.

Assumptions for the procedure. Let $S$ be the least element of $\mathcal{S}$
which has a vertex with an uncoloured incident edge. Call a vertex $x$
_pinned_ if it is a neighbour of some vertex $y\in S^{\prime}<S$ and
_unpinned_ otherwise (in particular, every vertex of $S$ is pinned). Notice
that each pinned vertex has at least one coloured back edge.

The following properties need to be satisfied before each execution of the
procedure:

  1. (A1)

every $S^{\prime}<S$ has all incident edges coloured, no other edges are
coloured,

  2. (A2)

the partial colouring defined so far fixes all $S^{\prime}<S$ pointwise with
respect to the group of automorphisms of $(G,r)$,

  3. (A3)

if $S^{\prime}\in\mathcal{S}\setminus\\{S_{1}\\}$ and each element of
$S^{\prime}$ has exactly $b$ coloured back edges, then
$|S^{\prime}|\leq\max\\{1,\Delta-b\\}$,

  4. (A4)

if a vertex $v$ has every incident edge coloured and all of them are red, then
either $v=r$ or $v$ is a unique neighbour of $r$ with all incident edges
coloured with red.

It is easy to verify that these assumptions are satisfied after the initial
colouring of the edges incident to $r$.

Procedure. Let $H$ be a component of the subgraph induced by the horizontal
edges of $S$. Notice that $H$ is a $\Delta^{\prime}$-regular graph for some
$\Delta^{\prime}<\Delta$. If $H\notin\mathcal{H}$, then we colour $H$ with a
distinguishing colouring using at most two colours, which exists by our
induction assumption. If $H\in\mathcal{H}$, then $H$ is traceable. We colour a
Hamiltonian path $P$ of $H$ with blue and we colour each remaining edges of
$H$ with red. This colouring breaks all the automorphisms of $H$ except these
which swap two endvertices of $P$. Notice that if $H\in\mathcal{H}$, then each
vertex in $H$ has an incident blue edge, so (A4) remains satisfied in that
case.

Since $|H|\geq h+1$ and each element of $S$ has exactly $b$ coloured back
edges, it follows from (A3) that there are at most $\frac{\Delta}{h+1}$
components of $G[S]$ if $S=S_{1}$ and at most $\frac{\Delta-b}{h+1}$
otherwise. Let $A$ be the set of automorphisms of $\operatorname{Aut}(G,r)$
which preserve the current colouring, and let $A^{\prime}$ be the set of these
automorphisms in $A$ which fix $S$.

If $f=0$, then we are done. If not, then we now colour forward edges. For each
vertex $x$ in $S$ we choose a number $i(x)$ between $0$ and $f$. We shall then
colour exactly $i(x)$ forward edges of $x$ with red, and the other ones with
blue. We need the following conditions on $i(x)$ in order to break the
remaining automorphisms inside $S$ and to guarantee (A3):

  1. (B1)

if $g$ is an isomorphism of $A$ which maps a component $H$ of $G[S]$ into a
different component of $G[S]$, then $g(i(x))\neq i(x)$ for some $x\in H$,

  2. (B2)

if $H\in\mathcal{H}$ and both $x$ and $y$ are endvertices of $P$, then
$i(x)\neq i(y)$,

  3. (B3)

if $f>1$, then only one vertex $x\in G[S]$ may have $i(x)=0$ and it may happen
only if $|S|\geq\Delta-1$ or $|S|=1$,

  4. (B4)

if $f>1$, then only one vertex $x\in G[S]$ may have $i(x)=f$ and it may happen
only if $|S|=\Delta$ and either $H=K_{1}$, or $H=K_{2}$ and $\Delta=4$,

  5. (B5)

if $f>1$ and there exists $x,y\in S$ such that $x$ and $y$ do not have the
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We now prove that we can always choose such $i(x)$ for every $x\in S$. Note
first that it follows from (A3) that the number of components in $G[S]$ is at
most $\frac{\Delta-b}{h+1}=\frac{f+h}{h+1}=1+\frac{f-1}{h+1}\leq f$ if $S\neq
S_{1}$ or at most $\frac{\Delta}{h+1}=\frac{f+h+1}{h+1}=1+\frac{f}{h+1}\leq
f+1$ if $S=S_{1}$. Furthermore, if there is $f+1$ components of $G[S]$, then
$h=0$ and $f=\Delta-1$, and if there are $f$ components of $G[S]$, then either
$h=0$, or $h=1$ and $f=2$. Hence for $f=1$ there exists only one component $H$
of $G[S]$. If $H\in\mathcal{H}$, then we assign 1 to one arbitrarily chosen
endvertex of the blue Hamiltonian path $P$ and 0 to each remaining vertex of
$H$. If $H\notin\mathcal{H}$, then we just assign 0 to each vertex of $H$. For
$f>1$ we select a component $H$ and put $i_{0}=\lceil\frac{f}{2}\rceil$ on all
the vertices except possibly one endvertex of the Hamiltonian path $P$ where
we put $i_{0}-1$. Note that $i_{0}-1>0$ unless $f=2$. Note that between any
two components there is exactly one isomorphism (if $H\not\in\mathcal{H}$) or
exactly two (otherwise). We then colour the other components so that all the
isomorphism between the components are broken and the endvertices of $P$ get
distinct colours. It is always possible, since we can pick a vertex other than
the previously chosen endvertex of $P$ and put all the other values of $i(x)$
on its equivalents in the other components. We however try to be more frugal,
keeping the labels as close to $i_{0}$ as possible. But even in the worst case
we are able to use any label $l\not\in\\{i_{0}-1,i_{0}\\}$ at most once, only
when the number of components is sufficiently high. Note that no vertex $x$ of
$H$ is assigned with $i(x)=f$ unless $f=\Delta-1$ and either $h=0$, or $h=1$
and $f=2$, which is equivalent to condition (B4). Similarly, none vertex $x$
of $H$ is assigned with $i(x)=0$ unless $|S|\geq\Delta-1$ or $|S|=1$.

Considering $x$ one by one we colour $i(x)$ forward edges of $x$ with red and
the remaining ones by blue. First we colour the edges incident to the vertices
$x$ with $i(x)=f$ and with $i(x)=0$.

Now we argue it is possible to colour the remaining forward edges so that
condition (A3) shall be satisfied in the next step. Consider some forward
neighbour of $S$ and its orbit $X$ with respect to $A^{\prime}$. The orbit $X$
has at most $\Delta-1$ elements. Moreover, by (B5), if $S=S_{1}$ and $z,t\in
S_{X}$ with $i(z)=f$, $i(t)=0$, then $|X|\leq\Delta-2$, if $S_{1}$ has two
vertices with distinct sets of forward neighbours, or $G=K_{\Delta,\Delta}$
(which satisfies the claim as a Hamiltonian graph) otherwise.

Each element of $X$ has the same set of neighbours $S_{X}$ in $S$. We colour
forward edges of vertices of $S_{X}$ in such a way as to split possibly the
most orbits. By splitting we mean that some two vertices in such an orbit get
incident edges of different colours to a common neighbour in $S$. Any given
vertex $s\in S$ has however a limited splitting ability, as each split
requires using two different colours. The number of orbits $s\in S$ can split
is therefore at most $\min\\{i(s),f-i(s)\\}$. Fortunately, the maximum number
of orbits to split is also limited – note that each such orbit must be non-
trivial and contain at least two vertices. So if we denote by
$X_{1},X_{2},\ldots,X_{l}$ where $X_{l}=X$ all the non-trivial orbits among
the forward neighbours of $s\in S_{X}$, then
$|X_{k}|+2(l-1)\leq|X_{1}|+|X_{2}|+\ldots+|X_{l}|\leq\Delta-1$ and
$|X_{k}|\leq\Delta-2l+1$ for each $k\in\\{1,\ldots,l\\}$. Let
$a_{k}=|N(X_{k})\cap S_{X}|$ and let $d_{k}\geq 0$ be the number of already
coloured back edges of any vertex in $X_{k}$, for $k=1,\ldots,l$.

If one of the $X_{k}$’s, say $X_{1}$, consists of exactly two vertices, then
the violation of the condition (A3) is possible only if $2>\max\\{1,\Delta-
a_{1}\\}$, which is equivalent to $a_{1}\geq\Delta-1$. Let $0\leq m\leq l$ be
the number of such orbits and assume that these are $X_{1},\ldots,X_{m}$ for
all $s\in S_{X}$. We remove the other orbits $X_{k}$ of order two from the
further consideration (we shall colour the edges to these orbits but without
any attempts to split them). By counting the vertices we obtain that
$2m+3(l-m)\leq f$.

Recall that any vertex $s\in S_{X}$ has $i(s)$ units of red colour and
$f-i(s)$ units of blue colour. Assume now that each vertex $s\in S_{X}$ splits
the maximum number of orbits, starting with $X_{1}$ (where $s$ splits one
largest remaining non-trivial orbit, if there is one) and continuing in
ascending order of indexes until it runs out of one of two colours. We show
that each of the orbits $X_{1},X_{2},\ldots,X_{l}$ is split sufficiently many
times.

Case 1. $m\geq 2$. We slightly change the order of colouring the vertices in
$S_{X}$ and make $m$ vertices with $i(s)<l$ or $i(s)>f-l$ first (or as many of
them as possible). Note that the first $m$ vertices necessarily split all the
orbits $X_{1},\ldots,X_{m}$. The number of the remaining vertices in $S_{X}$
which do not split $X_{k}$, $m<k\leq l$, i.e. such that $i(s)<l-m$ or
$i(s)>f-l+m$, is at most $2l-1-m$ for $S\neq S_{1}$ and at most $2l-m$ for
$S=S_{1}$ (as $f\geq 2m+3(l-m)\geq 3+3(l-m)$ implies that
$l-m\leq\frac{f}{3}-1<i_{0}-1$ for all $f>0$, therefore these labels have been
used at most once in $S$).

Case 2. $m\leq 1$. We do not modify the order of vertices and keep the
reasoning from Case 1 as if $m=0$. The only change is the justification that
$l<i_{0}-1$. Here it results from: $f\geq 2m+3(l-m)=3l-m\geq 3l-1\geq 2l+3$
hence $l\leq\frac{f}{3}-1<i_{0}-1$.

In both cases after colouring $S_{X}$, the orbit $X_{k}$ is split at least
$a_{k}-2l+1$ times (or $a_{k}-2l$ if $S=S_{1}$) and hence the largest orbit in
$X_{k}$ with respect to the obtained colouring has at most $\Delta-a_{k}$
vertices. But $a_{k}$ is exactly the number of coloured back edges of any
vertex in $X_{k}$. We then take any remaining $s\in S$ which has any incident
edges uncoloured and the orbits in the set of its forward neighbours and
repeat the above steps to colour the edges incident to $s$ and possibly some
other vertices in $S$. We repeat these steps until there are no such vertices
in $S$.

It remains to prove that the procedure assumptions are satisfied.

  1. (A1)

For each $s\in S$, we coloured all the edges incident to $s$ when we
considered that vertex. No other edges have been coloured.

  2. (A2)

If $S\neq S_{1}$ and $f=0$, then $S$ must be a singleton - otherwise by (A3)
for $S$ (which held before the execution of the procedure)
$|S|\leq\Delta-b=h<|S|$. If $S\neq S_{1}$ and $f\geq 1$, then we have broken
all the automorphisms of $G[S]$ except possibly the ones that mirror a
component of $G[S]$ or the ones that swap the components of $G[S]$. These
exceptions are broken by the choice of $i(s)$, $s\in S$. If $S=S_{1}$, then
$f\geq 1$ unless $G=K_{n}$ and the argument from the previous sentence
applies.

  3. (A3)

if $S^{\prime}\in\mathcal{S}\setminus\\{S_{1}\\}$ and each element of
$S^{\prime}$ has exactly $b$ coloured back edges, then
$|S^{\prime}|\leq\max\\{1,\Delta-b\\}$,

  4. (A4)

if a vertex $v$ has every incident edge coloured and all of them are red, then
either $v=r$ or $v$ is a unique neighbour of $r$ with all incident edges
coloured with red.

We need to make sure that there are at most two red vertices. Notice that if a
vertex $x$ has a non-zero number of forward edges (with respect to any
considered $\mathcal{S}$), then it has a forward blue edge unless $x=r$ or $x$
is a unique neighbour of $r$ with all incident edges coloured with red.
Therefore, it is enough to consider only these vertices $x$ for which $f=0$
for all considered $\mathcal{S}$. Consider the step of colouring edges
incident to $S$ when the last back edges of $x$ are coloured. Notice that at
most $\Delta-1$ back edges incident to $x$ can be coloured in one step unless
$S=S_{1}$ and $|S|=\Delta$. In that case, one of the vertices in $S$ has all
forward edges coloured with blue. In particular, the edge between this vertex
and $x$ is blue. Therefore, we can assume that $|S|<\Delta$. It follows that
$x$ is pinned. If $X$ forms a trivial orbit before colouring $S$, then it gets
a blue edge incident to the first considered element $s_{1}$ of $S$. Suppose
that $x$ does not form a trivial orbit before colouring $S$ and that $xs_{1}$
is not blue. Notice that $x$ is fixed before colouring the last of its back
edges $xs_{k}$. Therefore, the edge $xs_{i}$ is blue for some $i\leq k$. We
proved that $x$ and (possibly) $z$ are the only vertices with all incident
edges coloured with red.

∎

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