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arXiv · 2212.05239

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

Shenwei Huang
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Abstract

In this paper, we give an optimal $χ$-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 $χ(G)\le \lceil \frac{11}{9}ω(G) \rceil$. 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 $χ$-boundedness, Journal of Graph Theory 93, 503--552, 2020] combined with careful inductive arguments and a nontrivial use of the König theorem for bipartite matching.

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Shenwei Huang. 2022-12-10. The optimal $χ$-bound for $(P_7,C_4,C_5)$-free graphs. https://arxiv.org/abs/2212.05239

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