The number of distinguishing colorings of a Cartesian product graph

A (vertex, edge, total, proper, etc.) coloring of a graph $G$ is called distinguishing (or symmetry breaking) if no non-identity automorphism of $G$ preserves it, and the distinguishing number, denoted by $D(G)$, is the smallest number of colors required for vertex coloring to be distinguishing. This terminology has been introduced by Albertson and Collins [2] in 1996 and initiated many results and generalizations. The concept is however at least two decades older, but it was called asymmetric coloring, e.g. see [3] by Babai.

When vertex coloring of a graph $G$ is distinguishing, we say that this coloring breaks all the symmetries of $G$. A $k$-distinguishing coloring is a coloring that uses exactly $k$ colors. For a positive integer $d$, a graph $G$ is $d$-distinguishable if there exists a distinguishing vertex coloring with $d$ colors. The distinguishing number of some classes of graphs are as follows: $D(K_{n})=n$, $D(K_{n,n})=n+1$, $D(P_{n})=2$ for $n\geq 2$, $D(C_{3})=D(C_{4})=D(C_{5})=3$ while $D(C_{n})=2$ for $n\geq 6$ [2].

Because many graphs in applications appear to be a product of smaller graphs, one interesting natural question for every graph theoretical index is to consider it for product graphs, especially for Cartesian products. For example, Bogstad and Cowen [4] proved that for $k\geq 4$, every hypercube $Q_{k}$ of dimension $k$ is $2$-distinguishable. Moreover, Imrich and Klavžar [8] showed that the distinguishing number of Cartesian powers of a connected graph $G$ is equal to two except for $K_{2}^{2},K_{3}^{2},K_{2}^{3}$. Meanwhile, Imrich, Jerebic, and Klavžar [7] proved that Cartesian products of relatively prime graphs whose sizes are close to each other can be distinguished with a small number of colors.

Several indices, relevant to the distinguishing number, were introduced by Ahmadi, Alinaghipour, and Shekarriz [1]. Two colorings $c_{1}$ and $c_{2}$ of a graph $G$ are equivalent if there is an automorphism $\alpha$ of $G$ such that $c_{1}(v)=c_{2}(\alpha(v))$ for all $v\in V(G)$. The number of non-equivalent distinguishing colorings of a graph $G$ with $\{1,\ldots,k\}$ (as the set of admissible colors) is denoted by $\Phi_{k}(G)$, while the number of non-equivalent $k$-distinguishing colorings of a graph $G$ with $\{1,\ldots,k\}$ is denoted by $\varphi_{k}(G)$. Evidently, when $G$ has no distinguishing colorings with exactly $k$ colors, we have $\varphi_{k}(G)=0$. For a graph $G$, the distinguishing threshold, $\theta(G)$, is the minimum number $t$ such that for any $k\geq t$, any arbitrary $k$-coloring of $G$ is distinguishing [1].

Shekarriz et al. [10] alternatively defined the distinguishing threshold terms of $\mathrm{Aut}(G)$. For a non-identity automorphism $\alpha$, let $c(\alpha)$ be the number of cycles of $\alpha$ as a permutation and put $c(\text{\rm id})=0$. Then the distinguishing threshold of a graph $G$ is

To date, the distinguishing threshold has been studied for the Johnson graphs [10], the corona product, vertex-sum, rooted product, and lexicographic product [9]. It is shown to be useful to calculate the distinguishing number of disconnected graphs and the lexicographic product [1, 9]. In this paper, we consider the distinguishing threshold for the Cartesian products.

Here in this paper, we want to consider the following questions. Let $G$ and $H$ be two prime graphs. What can be said about $\theta(G\Box H)$ in terms of $\theta(G)$ and $\theta(H)$? Moreover, what can be said about $\varphi(G\Box H)$ or $\Phi_{k}(G\Box H)$ in terms of relevant indices of factors? If it is not possible to answer these questions rapidly, can we find an explicit formula for these indices when we limit our attention to paths and cycles?

Some preliminaries of the Cartesian product, distinguishing coloring and the distinguishing threshold, are introduced in Section 2. In Section 3, a generalization of Lemma 4.1 by Gorzkowska and Shekarriz [5] is presented. The lemma gives a necessary and sufficient condition on the coloring of the Cartesian product of graphs to be distinguishing coloring. Afterwards, we consider $\theta(G\Box H)$, $\Phi_{k}(G\Box H)$ and $\varphi_{k}(G\Box H)$ in Sections 4 and 5.

The material mentioned in this section is mostly a generalization of [5, Lemma 4.1]. Here, we need it two-sided, so that it can be used to calculate the number of distinguishing colorings of an arbitrary Cartesian product graph. To do so, we have altered some notions therein as follows.

Suppose that $G=G_{1}\Box G_{2}\Box\cdots\Box G_{k}$ for some $k\geq 2$ is a connected graph decomposed into a prime factorization, and $f$ is a total coloring. Note that a vertex or an edge coloring can be easily transformed into a total coloring (by giving all edges or vertices a fixed color). Therefore, $f$ can also be a vertex or an edge coloring.

For $i=1,\ldots,k$, let $V(G_{i})=\{1_{i},\ldots,m_{i}\}$ and for each $j=1,...,m$, consider

where $1_{r}$ is the first vertex of $G_{r}$ in our fixed ordering. It is a vertex of $G$ and we can speak of $G_{i}^{u_{j}}$ which is defined in Equation 2.3.

Meanwhile, for each $i$, the set of colors that we use are equivalent classes of colored quotients layers $Q_{i}^{v}$ under some equivalent relations we introduce here. Suppose that $\alpha\in\mathrm{Aut}(Q_{i})$. We define a map, namely $\varphi_{\alpha}$, from $Q_{i}^{u_{j}}$ onto $Q_{i}^{u_{t}}$, using $\alpha$ and our fixed ordering of $Q_{i}$, so that $\varphi_{\alpha}$ maps the vertex of $Q_{i}^{u_{j}}$ with the same ordering as $x\in Q_{i}$ onto the vertex of $Q_{i}^{u_{t}}$ with the same ordering as $\alpha(x)$. In this case, it is evident that $\varphi_{\alpha}$ is an isomorphism and we say it is a lifting of $\alpha$. Roughly speaking, the lifting of an automorphism produces an isomorphism from one copy of $Q_{i}$ onto another.

The coloring $f$ induces a (total) coloring on $Q_{i}^{u_{j}}$. This colored graph is denoted here by $\check{Q}_{i}^{u_{j}}=(Q_{i}^{u_{j}},f)$. For an automorphism $\alpha\in\operatorname{Aut}(Q_{i})$, we say that the color $\check{Q}_{i}^{u_{j}}$ is $\alpha$-equivalent to $\check{Q}_{i}^{u_{t}}$ if there is a (total) color-preserving isomorphism $\varphi:Q_{i}^{u_{j}}\longrightarrow Q_{i}^{u_{t}}$ which is a lifting of $\alpha$ or $\alpha^{-1}$.

Let $e=v_{i}w_{i}\in E(G_{i})$ and $\overline{Q_{i}^{e}}$ be the graph isomorphic to $Q_{i}$ constructed as follows: its vertex set consists of edges of $G$ of the form

for $x\in V(Q_{i})$ (i.e. $x_{j}\in V(G_{j})$, for $j\neq i$). If $x$ and $y$ are adjacent in $Q_{i}$, then vertices $(u_{i},x)(v_{i},x)$ and $(u_{i},y)(v_{i},y)$ are adjacent in $\overline{Q_{i}^{e}}$. Each vertex $(u_{i},x)(v_{i},x)$ of $\overline{Q_{i}^{e}}$ is an edge of $G$, so it is colored by the total coloring $f$. Therefore, $f$ induces a vertex coloring on $\overline{Q_{i}^{e}}$, and this colored graph is denoted here by $\hat{\overline{Q_{i}^{e}}}=(\overline{Q_{i}^{e}},f)$. Similarly, the colored graph $\hat{\overline{Q_{i}^{e}}}$ is $\alpha$-equivalent to $\hat{\overline{Q_{i}^{e^{\prime}}}}$ if there is a vertex-color-preserving isomorphism $\vartheta:\overline{Q_{i}^{e}}\longrightarrow\overline{Q_{i}^{e^{\prime}}}$ which is a lifting of $\alpha$ or $\alpha^{-1}$.

For each $i\in\{1,\ldots,k\}$, we can color vertices and edges of $G_{i}$ by colored graphs $Q_{i}$s as follows; let $G_{i}^{f}$ be the total coloring of $G_{i}$ in which each vertex $j\in V(G_{i})$ is colored by $\check{Q}_{i}^{u_{j}}$ and each edge $e=j{\ell}\in E(G_{i})$ is colored by $\hat{\overline{Q_{i}^{e}}}$. To distinguish this coloring from the similar coloring of [5], this total coloring of $G_{i}$ is called the holographic coloring of $G_{i}$ induced by $f$. Finally, two coloring $G_{i}^{f}$ and $G_{j}^{f}$ are equivalent if there is a total-color-preserving isomorphism from one to another. In this case we write $G_{i}^{f}\simeq G_{j}^{f}$

Now, we can articulate the desired statement.

As noted at the beginning of this section, Lemma 3.1 must also be considered true whenever $f$ is a vertex or edge coloring. However, when $f$ is a vertex coloring of $G$, it is redundant to color each edge $e$ of $G_{i}$ with $\hat{\overline{Q_{i}^{e}}}$, because all edges of $G_{i}$ will receive the same color.

It should also be noted that Lemma 3.1 remains true if the condition ii is true for each $\alpha\in\mathrm{Aut}^{F}(Q_{i})$ instead of for each $\alpha\in\mathrm{Aut}(Q_{i})$. As noted, $\mathrm{Aut}^{F}(Q_{i})$ is a subgroup of $\mathrm{Aut}(Q_{i})$ and the condition $G_{i}^{f}\not\simeq G_{j}^{f}$ for all $j\neq i$ makes it redundant to check the lemma for $\alpha\in\mathrm{Aut}(Q)\setminus\mathrm{Aut}^{F}(Q)$.

The following examples illustrate the holographic coloring and Lemma 3.1 interpretations.

Using Lemma 3.1 we are able to calculate the distinguishing threshold of the Cartesian product of prime graphs.

Theorem 2.2 implies that $\operatorname{Aut}(G^{k})\cong\mathrm{Sym}(k)\oplus\operatorname{Aut}(G)^{k}$. We use this fact in the proof of the following theorem.

Using the Cartesian prime factorization of a graph and Theorems 4.1 and 4.2, one can easily prove the next theorem, which concludes this section.

In this section, we calculate $\Phi_{k}(G\Box H)$ and $\varphi_{k}(G\Box H)$ when $G$ and $H$ are paths.

When the two factors are the same, calculations via inclusion-exclusion are a bit harder as there are automorphisms that exchange the factors. When the size of the automorphism group increases, it becomes hard to apply and validate an inclusion-exclusion argument. However, if we can classify similar states, we can easily use the argument. We do this in the proof of the following theorem.