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

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].Comment: 30 page