Minimum non-chromatic-$λ$-choosable graphs

For a multi-set $λ=\{k_1,k_2, \ldots, k_q\}$ of positive integers, let $k_λ = \sum_{i=1}^q k_i$. A $λ$-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 $λ$-choosable if $G$ is $L$-colourable for any $λ$-list assignment $L$ of $G$. The concept of $λ$-choosability puts $k$-colourability and $k$-choosability in the same framework: If $λ= \{k\}$, then $λ$-choosability is equivalent to $k$-choosability; if $λ$ consists of $k$ copies of $1$, then $λ$-choosability is equivalent to $k$-colourability. If $G$ is $λ$-choosable, then $G$ is $k_λ$-colourable. On the other hand, there are $k_λ$-colourable graphs that are not $λ$-choosable, provided that $λ$ contains an integer larger than $1$. Let $ϕ(λ)$ be the minimum number of vertices in a $k_λ$-colourable non-$λ$-choosable graph. This paper determines the value of $ϕ(λ)$ for all $λ$.

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