The optimal $\chi$-bound for $(P_7,C_4,C_5)$-free graphs

TL;DR

This paper establishes an optimal chromatic bound for a specific class of graphs defined by forbidden induced subgraphs, improving understanding of their coloring properties.

Contribution

It provides the first optimal $ ext{chi}$-binding function for $(P_7,C_4,C_5)$-free graphs, combining decomposition theorems and advanced combinatorial techniques.

Findings

Every $(P_7,C_4,C_5)$-free graph satisfies $ ext{chi}(G) extless= rac{11}{9} ext{omega}(G)$.

The proof uses a decomposition theorem and K"{o}nig's theorem for bipartite matching.

The result is proven to be optimal for this class of graphs.

Abstract

In this paper, we give an optimal $\chi$-binding function for the class of $(P_7,C_4,C_5)$-free graphs. We show that every $(P_7,C_4,C_5)$-free graph $G$ has $\chi(G)\le \lceil \frac{11}{9}\omega(G) \rceil$. To prove the result, we use a decomposition theorem obtained in [K. Cameron and S. Huang and I. Penev and V. Sivaraman, The class of $({P}_7,{C}_4,{C}_5)$-free graphs: Decomposition, algorithms, and $\chi$-boundedness, Journal of Graph Theory 93, 503--552, 2020] combined with careful inductive arguments and a nontrivial use of the K\"{o}nig theorem for bipartite matching.