Minimum non-chromatic-$\lambda$-choosable graphs

TL;DR

This paper characterizes the minimum size of graphs that are colorable with a certain number of colors but not list-colorable under a generalized framework called $oldsymbol{oldsymbol{oldsymbol{ extit{ extlambda}}}}$-choosability, for all such parameters.

Contribution

It determines the exact value of $oldsymbol{oldsymbol{oldsymbol{ extphi}}}(oldsymbol{ extlambda})$ for all $oldsymbol{ extlambda}$, extending the understanding of graph colorability versus list-colorability.

Findings

Calculated $oldsymbol{oldsymbol{oldsymbol{ extphi}}}(oldsymbol{ extlambda})$ for all $oldsymbol{ extlambda}$.

Unified framework connecting $k$-colorability and $k$-choosability.

Identified minimal non-$oldsymbol{ extlambda}$-choosable graphs for all parameters.

Abstract

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 list assignment $L$ of $G$ such that the colour set $\bigcup_{v \in V(G)}L(v)$ 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 $G$. The concept of $\lambda$-choosability puts $k$-colourability and $k$-choosability in the same framework: If $\lambda = \{k\}$, then $\lambda$-choosability is equivalent to $k$-choosability; if $\lambda$ consists of $k $ copies of $1$, then $\lambda$-choosability is equivalent to $k $-colourability. If $G$ is $\lambda$-choosable, then $G$ is $k_{\lambda}$-colourable. On the other hand, there are $k_{\lambda}$-colourable graphs that are not $\lambda$-choosable, provided that $\lambda$ contains an integer larger than $1$. Let $\phi(\lambda)$ be the minimum number of vertices in a $k_{\lambda}$-colourable non-$\lambda$-choosable graph. This paper determines the value of $\phi(\lambda)$ for all $\lambda$.