Linial for lists

Abstract

Linial’s famous color reduction algorithm reduces a given m-coloring of a graph with maximum degree varDelta to an O(varDelta ^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 beta , if every node has a list of size varOmega (beta ^2 (log beta +log log m + log log |{mathcal {C}}|)) from a color space {mathcal {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({text {deg}}+1)-list coloring algorithm from Barenboim et al. (PODC, pp 437–446, 2018) by slightly reducing the runtime to O(sqrt{varDelta log varDelta })+log ^* n and significantly reducing the message size (from varDelta ^{O(log ^* varDelta )} to roughly varDelta ). Our techniques are inspired by the local conflict coloring framework of Fraigniaud et al. (in: FOCS, pp 625–634, 2016).