Girth and $λ$-choosability of graphs

Abstract

Assume $ k $ is a positive integer, $ λ=\{k_1,k_2,...,k_q\} $ is a partition of $ k $ and $ G $ is a graph. A $λ$-assignment of $ G $ is a $ k $-assignment $ L $ of $ G $ such that the colour set $ \bigcup_{v\in V(G)} L(v) $ can be partitioned into $ q $ subsets $ C_1\cup C_2\cup\cdots\cup C_q $ and for each vertex $ v $ of $ G $, $ |L(v)\cap C_i|=k_i $. We say $ G $ is $λ$-choosable if for each $λ$-assignment $ L $ of $ G $, $ G $ is $ L $-colourable. In particular, if $ λ=\{k\} $, then $λ$-choosable is the same as $ k $-choosable, if $ λ=\{1, 1,...,1\} $, then $λ$-choosable is equivalent to $ k $-colourable. For the other partitions of $ k $ sandwiched between $ \{k\} $ and $ \{1, 1,...,1\} $ in terms of refinements, $λ$-choosability reveals a complex hierarchy of colourability of graphs. Assume $λ=\{k_1, \ldots, k_q\} $ is a partition of $ k $ and $λ' $ is a partition of $ k'\ge k $. We write $ λ\le λ' $ if there is a partition $λ''=\{k''_1, \ldots, k''_q\}$ of $k'$ with $k''_i \ge k_i$ for $i=1,2,\ldots, q$ and $λ'$ is a refinement of $λ''$. It follows from the definition that if $ λ\le λ' $, then every $λ$-choosable graph is $λ'$-choosable. It was proved in [X. Zhu, A refinement of choosability of graphs, J. Combin. Theory, Ser. B 141 (2020) 143 - 164] that the converse is also true. This paper strengthens this result and proves that for any $ λ\not\le λ' $, for any integer $g$, there exists a graph of girth at least $g$ which is $λ$-choosable but not $λ'$-choosable.