The chromatic number of ($P_{5}, K_{5}-e$)-free graphs

Yian Xu

The chromatic number of ($P_{5}, K_{5}-e$)-free graphs

Combinatorics 2023-04-11 v2

Authors: Yian Xu

Abstract

Let $G$ be a graph. We use $\chi(G)$ and $\omega(G)$ to denote the chromatic number and clique number of $G$ respectively. A $P_5$ is a path on 5 vertices. A family of graphs $\mathcal{G}$ is said to be {\it $\chi$ -bounded} if there exists some function $f$ such that $\chi(G)\leq f(\omega(G))$ for every $G\in\mathcal{G}$ . In this paper, we show that the family of $(P_5, K_5-e)$ -free graphs is $\chi$ -bounded by a linear function: $\chi(G)\leq \max\{13,\omega(G)+1\}$ .

Keywords

chromatic number clique number chi-bounded p5-free graphs k5-e-free graphs graph theory

Cite

@article{arxiv.2210.04682,
  title  = {The chromatic number of ($P_{5}, K_{5}-e$)-free graphs},
  author = {Yian Xu},
  journal= {arXiv preprint arXiv:2210.04682},
  year   = {2023}
}

Comments

This paper needs to be rewrote and reorganized. The last section might not be fully correct, it needs some further check

The chromatic number of ($P_{5}, K_{5}-e$)-free graphs

Abstract

Keywords

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