Xuding Zhu

A graph G is (1, 3)-colorable if its vertices can be partitioned into subsets $$V_1$$ V 1 and $$V_2$$ V 2 so that every vertex in $$G[V_1]$$ G [ V 1 ] has degree at most 1 and every vertex in $$G[V_2]$$ G [ V 2 ] has degree at most 3. We prove that every graph with maximum average degree at most 28/9 is (1, 3)-colorable.

Assume $G$ is a graph, $(v_1,\ldots,v_k)$ is a sequence of distinct vertices of $G$, and $(a_1,\ldots,a_k)$ is an integer sequence with $a_i \in \{1,2\}$. We say $G$ is \emph{$(a_1,\ldots,a_k)$-list extendable} (respectively, \emph{$(a_1,\ldots,a_k)$-AT extendable}) with respect to $(v_1,\ldots,v_k)$ if $G$ is $f$-choosable (respectively, $f$-AT), where $f(v_i)=a_i $ for $i \in \{1,\ldots, k\}$, and $f(v)=3$ for $v \in V(G) \setminus \{v_1,\ldots, v_k\}$. Hutchinson proved that if $G$ is an outerplanar graph, then $G$ is $(2,2)$-list extendable with respect to $(x,y)$ for any vertices $x,y$. We strengthen this result and prove that if $G$ is a $K_4$-minor-free graph, then $G$ is $(2,2)$-AT extendable with respect to $(x,y)$ for any vertices $x,y$. Then we characterize all triples $(x,y,z)$ of a $K_4$-minor-free graph $G$ for which $G$ is $(2,2,2)$-AT extendable (as well as $(2,2,2)$-list extendable) with respect to $(x,y,z)$. We also characterize the pairs $(x,y)$ of a $K_4$-minor-free graph $G$ for which $G$ is $(2,1)$-AT extendable (as well as $(2,1)$-list extendable) with respect to $(x,y)$. Moreover, we characterize all triples $(x,y,z)$ of a 3-colorable graph $G$ with its maximum average degree less than $\frac{14}{5}$ for which $G$ is $(2,2,2)$-AT extendable with respect to $(x,y,z)$.

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.