• Corpus ID: 239768186

On the chromatic number of a family of odd hole free graphs

@article{Song2021OnTC,
  title={On the chromatic number of a family of odd hole free graphs},
  author={Jialei Song and Baogang Xu},
  journal={ArXiv},
  year={2021},
  volume={abs/2110.12710}
}

• Jialei Song, Baogang Xu
• Published 25 October 2021
• Mathematics
• ArXiv

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.