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

TL;DR

This paper investigates the minimum size of graphs that are not $ ext{lambda}$-choosable, establishing bounds and exact values for various configurations, extending classical graph coloring results to more complex list coloring scenarios.

Contribution

It introduces bounds and exact values for the minimum size of non-$ ext{lambda}$-choosable graphs, generalizing known results in graph coloring to $ ext{lambda}$-list assignments.

Findings

Established bounds for $ ext{phi}( ext{lambda})$ depending on $ ext{lambda}$ properties.

Proved exact values of $ ext{phi}( ext{lambda})$ in special cases.

Extended classical coloring bounds to $ ext{lambda}$-choosability.

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 $2k_{\lambda}+2 \leqslant \phi(\lambda) \leqslant 2k_{\lambda}+3$.