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arXiv · 2106.08871

Polynomial $\chi$-binding functions for $t$-broom-free graphs

Xiaonan Liu·Joshua Schroeder·Zhiyu Wang·Xingxing Yu
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Abstract

For any positive integer $t$, a \emph{$t$-broom} is a graph obtained from $K_{1,t+1}$ by subdividing an edge once. In this paper, we show that, for graphs $G$ without induced $t$-brooms, we have $\chi(G) = o(\omega(G)^{t+1})$, where $\chi(G)$ and $\omega(G)$ are the chromatic number and clique number of $G$, respectively. When $t=2$, this answers a question of Schiermeyer and Randerath. Moreover, for $t=2$, we strengthen the bound on $\chi(G)$ to $7\omega(G)^2$, confirming a conjecture of Sivaraman. For $t\geq 3$ and \{$t$-broom, $K_{t,t}$\}-free graphs, we improve the bound to $o(\omega^{t})$.

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Xiaonan Liu, Joshua Schroeder, Zhiyu Wang, Xingxing Yu. 2021-06-16. Polynomial $\chi$-binding functions for $t$-broom-free graphs. https://arxiv.org/abs/2106.08871

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