Colouring $(P_r+P_s)$-Free Graphs

Abstract

The k-Colouring problem is to decide if the vertices of a graph can be coloured with at most k colours for a fixed integer k such that no two adjacent vertices are coloured alike. If each vertex u must be assigned a colour from a prescribed list L(u) ⊆ {1, . . . , k}, then we obtain the List k-Colouring problem. A graph G is H-free if G does not contain H as an induced subgraph. We continue an extensive study into the complexity of these two problems for H-free graphs. The graph Pr + Ps is the disjoint union of the r-vertex path Pr and the s-vertex path Ps. We prove that List 3-Colouring is polynomial-time solvable for (P2 + P5)-free graphs and for (P3 + P4)-free graphs. Combining our results with known results yields complete complexity classifications of 3-Colouring and List 3-Colouring on H-free graphs for all graphs H up to seven vertices. We also prove that 5-Colouring is NP-complete for (P3 + P5)-free graphs.

List-three-coloring graphs with no induced $P_6+rP_3$

“…, as required. This proves (9). By ( 6), ( 7) and ( 9), it remains to show that L is equivalent to P .…”

List 3-Coloring Graphs with No Induced $$P_6+rP_3$$

“…Lemma 5 Let t be an integer and let G be a P t -free graph, and let X, Y ⊆ V(G) be disjoint, where X is stable and every component of Y has size at most p. Let Z be a set of connected subsets of size q of Y, each of which has an attachment in X. Let H be a hypergraph with vertex set X and hyperedge set {X(Z) :…”