3-Coloring $C_4$ or $C_3$-free Diameter Two Graphs

TL;DR

This paper proves that 3-Coloring can be solved in polynomial time for certain classes of diameter two graphs that avoid specific cycles, extending previous results and including list coloring variants.

Contribution

It extends polynomial-time solvability of 3-Coloring to broader classes of cycle-free diameter two graphs, including all (C4,Cs)-free graphs and (C3,C7)-free graphs.

Findings

Polynomial-time algorithm for (C4,Cs)-free graphs for any constant s

Polynomial-time algorithm for (C3,C7)-free graphs

Results also apply to List 3-Coloring

Abstract

The question of whether 3-Coloring can be solved in polynomial-time for the diameter two graphs is a well-known open problem in the area of algorithmic graph theory. We study the problem restricted to graph classes that avoid cycles of given lengths as induced subgraphs. Martin et. al. [CIAC 2021] showed that the problem is polynomial-time solvable for $C_5$-free or $C_6$-free graphs, and, $(C_4,C_s)$-free graphs where $s \in \{3,7,8,9\}$. We extend their result proving that it is polynomial-time solvable for $(C_4,C_s)$-free graphs, for any constant $s$, and for $(C_3,C_7)$-free graphs. Our results also hold for the more general problem List 3-Colouring.