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arXiv · 2305.11757

Linear $\chi$-binding functions for $\{P_3\cup P_2, gem\}$-free graphs

Athmakoori Prashant·S. Francis Raj·M. Gokulnath
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Abstract

Finding families that admit a linear $\chi$-binding function is a problem that has interested researchers for a long time. Recently, the question of finding linear subfamilies of $2K_2$-free graphs has garnered much attention. In this paper, we are interested in finding a linear subfamily of a specific superclass of $2K_2$-free graphs, namely $(P_3\cup P_2)$-free graphs. We show that the class of $\{P_3\cup P_2,gem\}$-free graphs admits $f=2\omega$ as a linear $\chi$-binding function. Furthermore, we give examples to show that the optimal $\chi$-binding function $f^*\geq \left\lceil\frac{5\omega(G)}{4}\right\rceil$ for the class of $\{P_3\cup P_2, gem\}$-free graphs and that the $\chi$-binding function $f=2\omega$ is tight when $\omega=2$ and $3$.

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Athmakoori Prashant, S. Francis Raj, M. Gokulnath. 2023-05-19. Linear $\chi$-binding functions for $\{P_3\cup P_2, gem\}$-free graphs. https://arxiv.org/abs/2305.11757

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