Minimum Non-Chromatic- λ -Choosable Graphs

For a multiset $\lambda =\left\{k_1,k_2,\dots ,k_q\right\}$ 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 ${\bigcup }_{v\in V\left(G\right)}L\left(v\right)$ can be partitioned into the disjoint union $C_1\cup C_2\cup \cdots \cup C_q$ of $q$ sets so that for each $i$ and each vertex $v$ of $G$, $\mid L\left(v\right)\cap C_i\mid =k_i$. 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 a same framework: If $\lambda =\left\{k\right\}$, 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 $\varphi \left(\lambda \right)$ be the minimum number of vertices in a $k_{\lambda }$-colourable non-$\lambda$-choosable graph. This paper determines the value of $\varphi \left(\lambda \right)$ for all $\lambda$.