Girth and $\lambda$choosability of graphs
Abstract
Assume $ k $ is a positive integer, $ \lambda=\{k_1,k_2,...,k_q\} $ is a partition of $ k $ and $ G $ is a graph. A $\lambda$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 $\lambda$choosable if for each $\lambda$assignment $ L $ of $ G $, $ G $ is $ L $colourable. In particular, if $ \lambda=\{k\} $, then $\lambda$choosable is the same as $ k $choosable, if $ \lambda=\{1, 1,...,1\} $, then $\lambda$choosable is equivalent to $ k $colourable. For the other partitions of $ k $ sandwiched between $ \{k\} $ and $ \{1, 1,...,1\} $ in terms of refinements, $\lambda$choosability reveals a complex hierarchy of colourability of graphs. Assume $\lambda=\{k_1, \ldots, k_q\} $ is a partition of $ k $ and $\lambda' $ is a partition of $ k'\ge k $. We write $ \lambda\le \lambda' $ if there is a partition $\lambda''=\{k''_1, \ldots, k''_q\}$ of $k'$ with $k''_i \ge k_i$ for $i=1,2,\ldots, q$ and $\lambda'$ is a refinement of $\lambda''$. It follows from the definition that if $ \lambda\le \lambda' $, then every $\lambda$choosable graph is $\lambda'$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 $ \lambda\not\le \lambda' $, for any integer $g$, there exists a graph of girth at least $g$ which is $\lambda$choosable but not $\lambda'$choosable.
 Publication:

arXiv eprints
 Pub Date:
 September 2021
 arXiv:
 arXiv:2109.00776
 Bibcode:
 2021arXiv210900776G
 Keywords:

 Mathematics  Combinatorics
 EPrint:
 10 pages