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.
Highlights
Graph coloring is a notoriously known and well-studied concept in both graph theory and theoretical computer science
We will exploit the fact that once we find an induced P4, the vertices that are not adjacent to it must induce a P4-free graph
It is clear that we may determine in polynomial time whether an instance of list-3-coloring permits an application of a basic reduction, and perform the basic reduction, if available
Summary
Graph coloring is a notoriously known and well-studied concept in both graph theory and theoretical computer science. Algorithms for subclasses of Pt -free graphs, which avoid one or more additional induced subgraphs, usually cycles, have been studied They might be a first step in the attempt to settle the case of Pt -free graphs. The 3-coloring algorithm that we develop to prove Theorem 1 cannot be directly extended to solve the more general list-3-coloring problem since it uses the 3-coloring algorithm for perfect graphs to deal with graphs avoiding C7 and C9 Apart from this one case, the algorithm works with the more general setting of list-3-coloring. After several branching steps with polynomially many branches and suitable structural reductions of the original graph G, the algorithm will transform a 3-coloring instance of a (2P4, C5)-free graph G to a set of polynomially many heavily structured list-3-coloring instances These structured instances can be encoded by a 2-SAT formula, whose satisfiability is solvable in linear time [33]
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