Counterexamples to Hedetniemi's conjecture

TL;DR

This paper presents counterexamples to Hedetniemi's conjecture, showing that the chromatic number of the product of two graphs can be less than the minimum of their individual chromatic numbers.

Contribution

It provides the first known counterexamples to Hedetniemi's conjecture, challenging a long-standing assumption in graph theory.

Findings

Counterexamples where $ ext{chromatic}(G imes H) < ext{min}( ext{chromatic}(G), ext{chromatic}(H))$

Demonstrates the conjecture does not hold universally

Implications for graph coloring theory and related problems

Abstract

The chromatic number of $G\times H$ can be smaller than the minimum of the chromatic numbers of finite simple graphs $G$ and $H$.

Equations

Full text

The tensor product $G\times H$ of finite simple graphs $G$ and $H$ has vertex set $V(G)\times V(H)$, and pairs $(g,h)$ and $(g^{\prime},h^{\prime})$ are adjacent if and only if $\{g,g^{\prime}\}\in E(G)$ and $\{h,h^{\prime}\}\in E(H)$. One can easily see that $\chi(G\times H)\leqslant\chi(G)$ because a proper coloring $\Psi$ of the graph $G$ can be lifted to the coloring $(g,h)\to\Psi(g)$ of $G\times H$. Similarly, a proper coloring of $H$ leads to a proper coloring of $G\times H$ with the same number of colors, so we get

[TABLE]

The classical conjecture of S. T. Hedetniemi [8] posited the equality for all $G$ and $H$. More than 50 years have passed since the conjecture appeared, and it keeps attracting serious attention of researchers working in graph theory and combinatorics; we mention four exhaustive survey papers [9, 11, 13, 18] for more detailed information on the topic. Here, we briefly recall that Hedetniemi’s conjecture was proved in many special cases, including graphs with chromatic number at most four [3], graphs containing large cliques [1, 2, 16], circular graphs and products of cycles [15], and Kneser graphs and hypergraphs [6]. The conjecture gave an impetus to the study of multiplicative graphs, which remains remarkably active and important in its own right [5, 12, 17]. A generalization of Hedetniemi’s conjecture to fractional chromatic numbers turned out to be true [19], but the version with directed graphs is false [10], as well as the one with infinite chromatic numbers [7, 14]. We show that the inequality (H) can be strict for finite simple graphs.

A standard tool in the study of Hedetniemi’s conjecture is the concept of the exponential graph as introduced in [3]. Let $c$ be a positive integer, and let $\Gamma$ be a finite graph that we allow to contain loops; the graph $\mathcal{E}_{c}(\Gamma)$ has all mappings $V(\Gamma)\to\{1,\ldots,c\}$ as vertices, and two distinct mappings $\varphi,\psi$ are adjacent if, and only if, the condition $\varphi(x)\neq\psi(y)$ holds whenever $\{x,y\}\in E(\Gamma)$. The relevance of $\mathcal{E}_{c}(\Gamma)$ to the problem is easy to see because the graph $\Gamma\times\mathcal{E}_{c}(\Gamma)$ has the proper $c$-coloring $(h,\psi)\to\psi(h)$. The idea of our approach lies in the fact that the proper $c$-colorings of $\mathcal{E}_{c}(\Gamma)$ become quite well-behaved if the graph $\Gamma$ is fixed and $c$ gets large; let us proceed to technical details and exact statements. A basic result in [3] tells that the constant mappings form a $c$-clique in $\mathcal{E}_{c}(\Gamma)$, which means that these mappings get different colors in a proper $c$-coloring. So a relabeling of colors can turn any proper $c$-coloring $\Psi:\mathcal{E}_{c}(\Gamma)\to\{1,\ldots,c\}$ into a suited one, in which a color $i$ is assigned to the constant mapping sending every vertex of $\Gamma$ to $i$.

Assume that $I(u,b)$ is a large class, that is, it contains more than $n^{2}c^{n-2}$ elements, and consider an arbitrary mapping $\varphi_{b}\in\Psi^{-1}(b)$. If every element $\psi_{ub}$ of $I(u,b)$ admitted a vertex $u^{\prime}\neq u$ with $\psi_{ub}(u^{\prime})\in\operatorname{Im}\varphi_{b}$, then there would be at most $n^{2}$ ways to choose $u^{\prime}$ and $\psi_{ub}(u^{\prime})$, while the remaining $n-2$ vertices would contribute at most a factor of $c^{n-2}$. This contradicts the cardinality assumption on $I(u,b)$, so we can actually find a $\psi_{ub}\in I(u,b)$ under which $u$ is an only vertex taking the color $b$ and also $\operatorname{Im}\varphi_{b}\cap\operatorname{Im}\psi_{ub}=\{b\}$. In other words, the equality $\Psi(\varphi)=b$ cannot hold unless there is a vertex $w\in N(u)$ satisfying $\varphi(w)=b$.

If there is a vertex $v\in V(\Gamma)$ for which $I(v,b)$ is large for at least $c-\sqrt[n]{n^{3}c^{n-1}}$ colors $b$, then we are done. Conversely, we can define more than $n^{3}c^{n-1}$ mappings $\varphi:V(\Gamma)\to\{1,\ldots,c\}$ for which the value of $\varphi$ on a vertex $w$ does not equal those colors $b$ for which $I(w,b)$ is large. None of these mappings belongs to a large class $I(u,b)$, but the non-large classes are too small to cover all of them. ∎

Now we are ready to proceed with counterexamples. For a simple graph $G$, we define the graph $\Gamma_{G}$ by adding the loops to all the vertices, and the strong product $G\boxtimes K_{q}$ as the graph with vertex set $V(G)\times\{1,\ldots,q\}$ and edges between $(u,i)$ and $(v,j)$ when, and only when, $\{u,v\}\in E(G)$ or $(u=v)\&(i\neq j)$.

(1.1) $\mu_{t}(g,i)=i$ for all $g\in V(G)$ satisfying $\operatorname{dist}(v,g)\in\{0,2\}$,

(1.2) $\mu_{t}(g,i)=q+i$ for all $g\in V(G)$ satisfying $\operatorname{dist}(v,g)=1$,

(1.3) $\mu_{t}(g,i)=t$ for all $g\in V(G)$ satisfying $\operatorname{dist}(v,g)\geqslant 3$.

Due to the assumption on the girth of $G$, no pair of vertices defined in (1.1) and (1.2) can be adjacent in $G\boxtimes K_{q}$ and monochromatic at the same time; the condition (1.3) uses different colors for different $t$, and these colors are also different from those of the neighboring vertices dealt with in (1.1). Therefore, $\mathcal{M}$ is indeed a clique and requires $c-q\geqslant 2.1q$ colors. Using the pigeonhole principle, one finds a $\tau\in\{q+1,\ldots,c\}$ such that $\Lambda(\mu_{\tau})\notin\{1,\ldots,2q\}$, and due to Observation 1 we have $\tau=\Lambda(\mu_{\tau})$. Further, it is only $o(q)$ classes that are not $v$-robust with respect to $\Psi$ in the terminology of Claim 2, so we can find a $v$-robust class $\sigma\notin\{1,\ldots,2q,\tau\}$. Finally, we note that the mapping $\nu:G\boxtimes K_{q}\to\{1,\ldots,c\}$ defined as, for all $i$,

(2.1) $\nu(g,i)=\tau$ for all $g\in V(G)$ in the closed neighborhood $N(v)$,

(2.2) $\nu(g,i)=\sigma$ for all $g\in V(G)$ satisfying $\operatorname{dist}(v,g)\geqslant 2$,

is adjacent to $\mu_{\tau}$ in $\mathcal{E}_{c}(G\boxtimes K_{q})$. Since $\sigma$ is $v$-robust, we cannot have $\Lambda(\nu)=\sigma$ by Lemma 2, but rather we have $\Lambda(\nu)=\tau$ according to Observation 1. So we have $\Lambda(\nu)=\Lambda(\mu_{\tau})$, which is a contradiction. ∎

The classical paper [4] proves the existence of graphs with arbitrarily large girth and fractional chromatic number; so we can find a graph $G$ of girth at least $6$ that satisfies $\chi_{f}(G)>3.1$. We set $c=\lceil 3.1q\rceil$ and pass to sufficiently large $q$; we immediately get $\chi(G\boxtimes K_{q})\geqslant q\cdot\chi_{f}(G)>c$ and also $\chi\left(\mathcal{E}_{c}(G\boxtimes K_{q})\right)>c$ by Claim 3. The equality $\chi\left((G\boxtimes K_{q})\times\mathcal{E}_{c}(G\boxtimes K_{q})\right)=c$ follows by standard theory [3] as the mapping $(u,\varphi)\to\varphi(u)$ is a proper $c$-coloring of any graph of the form $\Gamma\times\mathcal{E}_{c}(\Gamma)$.