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}, 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 is the disjoint union of the r-vertex path P_r and the s-vertex path P_s. We prove that List 3-Colouring is polynomial-time solvable for (P_2+P_5)-free graphs and for (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.
Highlights
Graph colouring is a popular concept in Computer Science and Mathematics due to a wide range of practical and theoretical applications, as evidenced by numerous surveys and books on graph colouring and many of its variants
If the graph G of an instance (G, L) of List 3-Colouring is P7-free, we can use the aforementioned result of Bonomo et al [3]
We find that the vertex w created by Rule 12 must be in V(P) ∪ V(P ), as otherwise, P + P was already an induced subgraph of G before Rule 12 was applied
Summary
Graph colouring is a popular concept in Computer Science and Mathematics due to a wide range of practical and theoretical applications, as evidenced by numerous surveys and books on graph colouring and many of its variants (see, for example, [1, 6, 15, 23, 26, 30, 32, 34]). Project UNCE/SCI/004), and by the Project GAUK 1277018. An extended abstract of this paper has appeared in the proceedings of ISAAC 2018 [24]
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