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

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 ∈{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.

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