Colouring (Pr + Ps)-Free Graphs
Tereza Klimošová, Josef Malík, Tomáš Masařík, Jana Novotná, Daniël Paulusma, Veronika Slívová · Algorithmica · 2020
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 $$L(u)\subseteq \{1,\ldots ,k\},$$ 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 $$P_r+P_s$$ P r + P s is the disjoint union of the r-vertex path $$P_r$$ P r and the s-vertex path $$P_s.$$ P s . We prove that List 3-Colouring is polynomial-time solvable for $$(P_2+P_5)$$ ( P 2 + P 5 ) -free graphs and for $$(P_3+P_4)$$ ( P 3 + P 4 ) -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.