An asymptotically tight bound on the adaptable chromatic number

The adaptable chromatic number of a multigraph G, denoted χa(G), is the smallest integer k such that every edge labeling of G from [k] = {1, 2, · · · , k} permits a vertex coloring of G from [k] such that no edge e = uv has c(e) = c(u) = c(v). Hell and Zhu proved that for any multigraph G with maximum degree ∆, the adaptable chromatic number is at most lp e(2∆− 1) m . We strengthen this to the asymptotically best possible bound of (1 + o(1)) √ ∆.