Linial for lists

Abstract

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