On 3-Coloring of $(2P_4,C_5)$-Free Graphs

TL;DR

This paper proves that the 3-coloring problem can be solved efficiently in polynomial time for graphs that do not contain two disjoint paths of four vertices and a five-cycle as induced subgraphs.

Contribution

It establishes the polynomial-time solvability of 3-coloring for the specific class of (2P4,C5)-free graphs, addressing an open case in graph coloring complexity.

Findings

3-coloring is polynomial-time solvable for (2P4,C5)-free graphs.

Addresses an open problem in the complexity of coloring hereditary graph classes.

Expands understanding of coloring complexity for graphs with forbidden induced subgraphs.

Abstract

The 3-coloring of hereditary graph classes has been a deeply-researched problem in the last decade. A hereditary graph class is characterized by a (possibly infinite) list of minimal forbidden induced subgraphs $H_1,H_2,\ldots$; the graphs in the class are called $(H_1,H_2,\ldots)$-free. The complexity of 3-coloring is far from being understood, even for classes defined by a few small forbidden induced subgraphs. For $H$-free graphs, the complexity is settled for any $H$ on up to seven vertices. There are only two unsolved cases on eight vertices, namely $2P_4$ and $P_8$. For $P_8$-free graphs, some partial results are known, but to the best of our knowledge, $2P_4$-free graphs have not been explored yet. In this paper, we show that the 3-coloring problem is polynomial-time solvable on $(2P_4,C_5)$-free graphs.