Linial for lists

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