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

TL;DR

This paper establishes an upper bound on the adaptable choosability of multigraphs with no triangles or quadrilaterals, linking it to maximum degree and extending to conflict choosability under certain conditions.

Contribution

It provides a new upper bound for adaptable and conflict choosability in multigraphs lacking small cycles, advancing understanding of graph coloring constraints.

Findings

Bound: $ ext{ch}_a(G) \, \leq \, (2\sqrt{2}+o(1))\sqrt{\Delta/\ln\Delta}$ for such graphs.

Extension of the bound to conflict choosability under natural restrictions.

Improves theoretical understanding of coloring properties in cycle-restricted multigraphs.

Abstract

The adaptable choosability of a multigraph $G$, denoted $\mathrm{ch}_a(G)$, is the smallest integer $k$ such that any edge labelling, $\tau$, of $G$ and any assignment of lists of size $k$ to the vertices of $G$ permits a list colouring, $\sigma$, of $G$ such that there is no edge $e = uv$ where $\tau(e) = \sigma(u) = \sigma(v)$. Here we show that for a multigraph $G$ with maximum degree $\Delta$ and no cycles of length 3 or 4, $\mathrm{ch}_a(G) \leq (2\sqrt{2}+o(1))\sqrt{\Delta/\ln\Delta}$. Under natural restrictions we can show that the same bound holds for the conflict choosability of $G$, which is a closely related parameter defined by Dvo\v{r}\'ak, Esperet, Kang and Ozeki [arXiv:1803.10962].