2021/10/25 by Jialei Song, Song, Jialei, Baogang Xu +1
Computer Science · Mathematics · #Advanced Graph Theory Research #Limits and Structures in Graph Theory #Graph Labeling and Dimension Problems
paper · pdf · doi:10.48550/arxiv.2110.12710
A hole is an induced cycle of length at least 4, and an odd hole is a hole of odd length. A full house is a graph composed by a vertex adjacent to both ends of an edge in K4 . Let H be the complement of a cycle on 7 vertices. Chudnovsky et al [6] proved that every (odd hole, K4)-free graph is 4-colorable and is 3-colorable if it does not has H as an induced subgraph. In this paper, we use the proving technique of Chudnovsky et al to generalize this conclusion to (odd hole, full house)-free graphs, and prove that for (odd hole, full house)-free graph G, χ(G)≤ ω(G)+1, and the equality holds if and only if ω(G)=3 and G has H as an induced subgraph.