A survey on Hedetniemi's conjecture

TL;DR

This survey captures the current state of Hedetniemi's conjecture by contrasting the cases where Hedetniemi-type equalities hold for homomorphism-monotone invariants (notably $\chi_f$, $\chi_v$, and $\chi_{sv}$) with the fundamental counterexamples to the original conjecture that sparked a broad research program. It surveys a diverse toolkit—linear programming duality, semidefinite programming, topological methods via the box complex (index/coindex), and categorical constructions like exponential graphs—to explain when equality can be established and when it cannot. The paper also coordinates the multiplicativity paradigm for graphs and digraphs, the role of the Poljak–Rödl function $f(n)$, and recent refinements that produce smaller and simpler counterexamples, while highlighting open questions about growth, limits, and the reach of these techniques. Overall, it charts a landscape where several invariants exhibit Hedetniemi-type behavior, yet the chromatic-number boundary remains decisively broken, inviting further exploration of topological, algebraic, and combinatorial approaches.

Abstract

In 1966, Hedetniemi conjectured that for any positive integer $n$ and graphs $G$ and $H$, if neither $G$ nor $H$ is $n$-colourable, then $G \times H$ is not $n$-colourable. This conjecture has received significant attention over the past half century, and was disproved by Shitov in 2019. Shitov's proof shows that Hedetniemi's conjecture fails for sufficiently large $n$. Shortly after Shitov's result, smaller counterexamples were found in a series of papers, and it is now known that Hedetniemi's conjecture fails for all $n \ge 4$, and holds for $n \le 3$. Hedetniemi's conjecture has inspired extensive research, and many related problems remain open. This paper surveys the results and problems associated with the conjecture, and explains the ideas used in finding counterexamples.

Figures (2)

• Figure 1: A non-$n$-colourable subgraph $H$ of $K_n^G$ for Theorem \ref{['thm-Zhu']}
• Figure 2: A non-$4$-colourable subgraph $H$ of $K_4^{\Omega_{13}(K_8)}$, where a vertex labeled $\{a\}$ indicates the constant mapping ${\rm const}_a$, a vertex labeled $\{ab\}$ indicates a mapping $\sigma_{a,b}^{\ell, S}$ for some $\ell \in \{0,1,\ldots, 6\}$ and $S \subseteq \{1,2,3,4\}$, and a vertex labeled $\{abc\}$ indicates a mapping $\tau_{a,b,}^{\ell, S, T}$ for some $\ell \in \{0,1,\ldots, 6\}$ and $S, T \subseteq \{1,2,3,4\}$. The labels of vertices also indicate allowed colors, as in list coloring of the graph.

Theorems & Definitions (39)

• Conjecture 1.1
• Definition 2.1
• Definition 2.2
• Theorem 2.4
• Theorem 2.5
• Definition 2.6
• Theorem 2.7
• Theorem 2.8
• Definition 2.9
• Theorem 2.10