A refinement of choosability of graphs

Abstract

Assume $k$ is a positive integer, $\lambda=\{k_1, k_2, \ldots, k_q\}$ is a partition of $k$ and $G$ is a graph. A $\lambda$-list assignment of $G$ is a $k$-list assignment $L$ of $G$ such that the colour set $\cup_{v\in V(G)}L(v)$ can be partitioned into $q$ subsets $C_1 \cup C_2 \ldots \cup C_q$ and for each vertex $v$ of $G$, $|L(v) \cap C_i| \ge k_i$. We say $G$ is $\lambda$-choosable if for each $\lambda$-list assignment $L$ of $G$, $G$ is $L$-colourable. It follows from the definition that if $\lambda =\{k\}$, then $\lambda$-choosable is the same as $k$-choosable, if $\lambda =\{1,1,\ldots, 1\}$, then $\lambda$-choosable is equivalent to $k$-colourable. For the other partitions of $k$ sandwiched between $\{k\}$ and $\{1,1,\ldots, 1\}$ in terms of refinements, $\lambda$-choosability reveals a complex hierarchy of colourability of graphs. We prove that for two partitions $\lambda, \lambda'$ of $k$, every $\lambda$-choosable graph is $\lambda'$-choosable if and only if $\lambda'$ is a refinement of $\lambda$. Then we concentrate on $\lambda$-choosability of planar graphs for partitions $\lambda$ of $4$. Several conjectures concerning colouring of generalized signed planar graphs are proposed and relations between these conjectures and list colouring conjectures for planar graphs are explored. In particular, it is proved that a conjecture of K\"{u}ndgen and Ramamurthi on list colouring of planar graphs is implied by the conjecture that every planar graph is $\{2,2\}$-choosable, and also implied by the conjecture of M\'{a}\v{c}ajov\'{a}, Raspaud and \v{S}koviera which asserts that every planar graph is signed MRS-$4$-colourable, and that a conjecture of Kang and Steffen asserting that every planar graph is signed KS-$4$-colourable implies that every planar graph is $\{1,1,2\}$-choosable.