Linial for lists

Linial’s famous color reduction algorithm reduces a given m -coloring of a graph with maximum degree to an O( ^2log m) -coloring, in a single round in the LOCAL model. We give a similar result when nodes are restricted to choose their color from a list of allowed colors: given an m -coloring in a directed graph of maximum outdegree β , if every node has a list of size (β ^2 (logβ +loglog m + loglog |𝒞|)) from a color space 𝒞 then they can select a color in two rounds in the LOCAL model. Moreover, the communication of a node essentially consists of sending its list to the neighbors. This is obtained as part of a framework that also contains Linial’s color reduction (with an alternative proof) as a special case. Our result also leads to a defective list coloring algorithm. As a corollary, we improve the state-of-the-art truly local (deg+1) -list coloring algorithm from Barenboim et al. (PODC, pp 437–446, 2018) by slightly reducing the runtime to O(√(log))+log ^* n and significantly reducing the message size (from ^O(log ^* ) to roughly ). Our techniques are inspired by the local conflict coloring framework of Fraigniaud et al. (in: FOCS, pp 625–634, 2016).