Adaptable and conflict colouring multigraphs with no cycles of length three or four

Abstract

AbstractThe adaptable choosability of a multigraph , denoted , is the smallest integer such that any edge labelling, , of and any assignment of lists of size to the vertices of permits a list colouring, , of such that there is no edge where . Here we show that for a multigraph with maximum degree and no cycles of length 3 or 4, . Under natural restrictions we can show that the same bound holds for the conflict choosability of , which is a closely related parameter recently defined by Dvořák, Esperet, Kang and Ozeki.