Chromatic $\lambda$-choosable and $\lambda$-paintable graphs

Abstract

Let $\phi(k)$ be the minimum number of vertices in a non-$k$-choosable $k$-chromatic graph. The Ohba conjecture, confirmed by Noel, Reed and Wu, asserts that $\phi(k) \ge 2k+2$. This bound is tight if $k$ is even. If $k$ is odd, then it is known that $\phi(k) \le 2k+3$ and it is conjectured by Noel that $\phi(k) = 2k+3$. For a multi-set $\lambda=\{k_1,k_2, \ldots, k_q\}$ of positive integers, let $k_{\lambda} = \sum_{i=1}^q k_i$. A $\lambda$-list assignment of $G$ is a $k_{\lambda}$-list assignment $L$ for which the colour set $\cup_{v \in V(G)}L(v)$ can be partitioned into the disjoint union $C_1 \cup C_2 \cup \ldots \cup C_q$ of $q$ sets so that for each $i$ and each vertex $v$ of $G$, $|L(v) \cap C_i| \ge k_i$. We say $G$ is $\lambda$-choosable if $G$ is $L$-colourable for any $\lambda$-list assignment $L$ of $G$. Let $\phi(\lambda )$ be the minimum number of vertices in a non-$\lambda$-choosable $k_{\lambda}$-chromatic graph. Let $1_{\lambda}$ be the multiplicity of $1$ in $\lambda$, and let $o_{\lambda}$ be the number of elements in $\lambda$ that are odd integers. We prove that if $1_{\lambda} \ne k_{\lambda}$, then $2k_{\lambda}+1_{\lambda}+2 \leqslant \phi(\lambda ) \leqslant 2k_\lambda+ o_\lambda +2$. In particular, if $1_{\lambda}=o_{\lambda}=t$, i.e. $\lambda$ contains no odd integer greater than $1$, then $\phi(\lambda ) = 2k_{\lambda}+t+2$. We also prove that $\phi(\lambda) \leqslant 2k_{\lambda}+5 1_{\lambda}+3$. In particular, if $1_{\lambda}=0$, then $2k_{\lambda}+2 \leqslant \phi(\lambda) \leqslant 2k_{\lambda}+3$.