Minimal obstructions to C_5-coloring in hereditary graph classes

For graphs G and H, an H-coloring of G is an edge-preserving mapping from V(G) to V(H). Note that if H is the triangle, then H-colorings are equivalent to 3-colorings. In this paper we are interested in the case that H is the five-vertex cycle C_5. A minimal obstruction to C_5-coloring is a graph that does not have a C_5-coloring, but every proper induced subgraph thereof has a C_5-coloring. In this paper we are interested in minimal obstructions to C_5-coloring in F-free graphs, i.e., graphs that exclude some fixed graph F as an induced subgraph. Let P_t denote the path on t vertices, and let S_a,b,c denote the graph obtained from paths P_a+1,P_b+1,P_c+1 by identifying one of their endvertices. We show that there is only a finite number of minimal obstructions to C_5-coloring among F-free graphs, where F ∈{ P_8, S_2,2,1, S_3,1,1} and explicitly determine all such obstructions. This extends the results of Kamiński and Pstrucha [Discr. Appl. Math. 261, 2019] who proved that there is only a finite number of P_7-free minimal obstructions to C_5-coloring, and of Dębski et al. [ISAAC 2022 Proc.] who showed that the triangle is the unique S_2,1,1-free minimal obstruction to C_5-coloring. We complement our results with a construction of an infinite family of minimal obstructions to C_5-coloring, which are simultaneously P_13-free and S_2,2,2-free. We also discuss infinite families of F-free minimal obstructions to H-coloring for other graphs H.