Codegree Thresholds for $λ$-Choosability of Graphs

Abstract

Let $λ=\{k_1,\ldots,k_q\}$ be a partition, and let $|λ|=k_1+\cdots+k_q$. A $|λ|$-list assignment $L$ of a graph $G$ is a $λ$-assignment if its color set can be partitioned into $q$ disjoint sets $X_1,\ldots,X_q$ such that $|L(v)\cap X_i|=k_i$ for every vertex $v$ and every $i\in[q]$. This notion, introduced by Zhu [J. Combin. Theory Ser. B, 2020], puts ordinary coloring and list coloring in the same framework. A theorem of Alon [Random Structures Algorithms, 2000] states that every graph with minimum degree $d$ has choice number at least $(1/2-o(1))\log_2d$. Saxton and Thomason [Invent. Math., 2015] later used the hypergraph container method to replace $1/2$ by the sharp constant $1$. It is natural to ask whether a similar phenomenon holds for every fixed partition $λ$. Minimum degree alone is not sufficient: balanced complete bipartite graphs have arbitrarily large minimum degree but are always $\{1,1\}$-choosable. We show that the appropriate replacement is the minimum $q$-codegree, defined for $|V(G)|\geq q$ by $δ_q(G)=\min\{|N_G(S)|:S\subseteq V(G),\,|S|=q\}$. More precisely, for every partition $λ$ there exists an integer $d$ such that every graph $G$ with $δ_q(G)\geq d$ is not $λ$-choosable. Let $f(λ)$ be the least such $d$. For every fixed $q$, we prove $f(λ)\leq2^{(2q+o(1))|λ|}$ as $|λ|\to\infty$, while $f(λ)\geq(q+1)^{-1}(1+1/q)^{|λ|}$ for every $λ$. For the partition $\{k,\ldots,k\}$ with $q$ equal parts, we determine the threshold asymptotically: $f(\{k,\ldots,k\})=ρ_q^{-(1+o(1))k}$ as $k\to\infty$, where $ρ_q$ is the unique $x\in(0,1)$ satisfying $x=(1-x)^q$. When $q=1$, our result implies $\operatorname{ch}(G)\geq(1-o(1))\log_2δ(G)$.