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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, weuse a decomposition theorem obtained in [K. Cameron and S. Huang and I. Penevand 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 useof the K\"{o}nig theorem for bipartite matching.

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