Colouring Generalized Claw-Free Graphs and Graphs of Large Girth: Bounding the Diameter

TL;DR

This paper investigates the complexity of the k-Colouring problem on H-free graphs with large girth and bounded diameter, exploring how these constraints influence computational difficulty.

Contribution

It extends existing NP-completeness results by analyzing the impact of bounded diameter on coloring problems in graphs with large girth.

Findings

NP-completeness persists for certain classes of graphs with large girth and bounded diameter.

Bounded diameter can alter the complexity landscape of graph coloring problems.

Results provide new insights into the interplay between graph structure and coloring complexity.

Abstract

For a fixed integer, the $k$-Colouring problem is to decide if the vertices of a graph can be coloured with at most $k$ colours for an integer $k$, such that no two adjacent vertices are coloured alike. A graph $G$ is $H$-free if $G$ does not contain $H$ as an induced subgraph. It is known that for all $k\geq 3$, the $k$-Colouring problem is NP-complete for $H$-free graphs if $H$ contains an induced claw or cycle. The case where $H$ contains a cycle follows from the known result that the problem is NP-complete even for graphs of arbitrarily large fixed girth. We examine to what extent the situation may change if in addition the input graph has bounded diameter.