List k-colouring P-free graphs: a mim-width perspective

Abstract

A colouring of a graph G = (V, E) is a mapping c : V → {1, 2, . . .} such that c(u) 6= 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 1,s 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 (DM 2020) proved that List 3-Colouring is polynomialtime solvable for (K1 1,s, 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 1,s, Pt)-free graphs has bounded mim-width and that a corresponding branch decomposition is “quickly computable” for these graphs.