List k-colouring P_t-free graphs: A Mim-width perspective

A colouring of a graph G=(V,E) is a mapping c:V→{1,2,…} such that c(u)≠c(v) for every two adjacent vertices u and v of G. The LIST k-COLOURING problem is to decide whether a graph G=(V,E) with a list L(u)⊆{1,…,k} for each u∈V has a colouring c such that c(u)∈L(u) for every u∈V. Let Pt be the path on t vertices and let K1,s1 be the graph obtained from the (s+1)-vertex star K1,s by subdividing each of its edges exactly once. Recently, Chudnovsky, Spirkl and Zhong (Discrete Math. 2020) proved that LIST 3-COLOURING is polynomial-time solvable for (K1,s1,Pt)-free graphs for every t≥1 and s≥1. We generalize their result to LIST k-COLOURING for every k≥1. Our result also generalizes the known result that for every k≥1 and s≥0, LIST k-COLOURING is polynomial-time solvable for (sP1+P5)-free graphs, which was proven for s=0 by Hoàng, Kamiński, Lozin, Sawada and Shu (Algorithmica 2010) and for every s≥1 by Couturier, Golovach, Kratsch and Paulusma (Algorithmica 2015). We show our result by proving boundedness of an underlying width parameter. Namely, we show that for every k≥1, s≥1, t≥1, the class of (Kk,K1,s1,Pt)-free graphs has bounded mim-width and that a corresponding branch decomposition is “quickly computable” for these graphs.