Girth and λ $\lambda $‐choosability of graphs

Abstract

AbstractAssume is a positive integer, is a partition of and is a graph. A ‐assignment of is a ‐assignment of such that the colour set can be partitioned into subsets and for each vertex of , . We say is ‐choosable if for each ‐assignment of , is ‐colourable. In particular, if , then ‐choosable is the same as ‐choosable, and if , then ‐choosable is equivalent to ‐colourable. For the other partitions of sandwiched between and in terms of refinements, ‐choosability reveals a complex hierarchy of colourability of graphs. Assume is a partition of and is a partition of . We write if there is a partition of with for and is a refinement of . It follows from the definition that if , then every ‐choosable graph is ‐choosable. It was proved in Zhu that the converse is also true. This paper strengthens this result and proves that for any , for any integer , there exists a graph of girth at least which is ‐choosable but not ‐choosable.