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

The adaptable choosability of a multigraph G $G$ , denoted ch ( ) ${\text{ch}}_{a}(G)$ is the smallest integer k $k$ such that any edge labelling, τ $\tau $ and assignment lists size to vertices permits list colouring, σ $\sigma there no e = u v $e=uv$ where (e)=\sigma (u)=\sigma (v)$ . Here we show for with maximum degree Δ ${\rm{\Delta }}$ cycles length 3 or 4, ≤ 2 + o 1 ∕ ln ${\text{ch}}_{a}(G)\,\le (2\sqrt{2}+o(1))\sqrt{{\rm{\Delta }}\unicode{x02215}\mathrm{ln}\unicode{x0200A}{\rm{\Delta }}}$ Under natural restrictions can same bound holds conflict which closely related parameter recently defined by Dvořák, Esperet, Kang Ozeki.