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The adaptable choosability of a multigraph $G$, denoted $\mathrm{ch}_a(G)$,is the smallest integer $k$ such that any edge labelling, $\tau$, of $G$ andany assignment of lists of size $k$ to the vertices of $G$ permits a listcolouring, $\sigma$, 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 maximumdegree $\Delta$ 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 canshow that the same bound holds for the conflict choosability of $G$, which is aclosely related parameter defined by Dvo\v{r}\'ak, Esperet, Kang and Ozeki[arXiv:1803.10962].

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