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

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.