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arXiv · 2607.15071

Two problems on booksize and triangular edges in Nosal graphs

Xinghui Zhao·Lihua You·Jing Zeng·Xiaoxue Zhang
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Abstract

A graph $G$ with $m$ edges is said to be a Nosal graph if $\rho(G)>\sqrt{m}$. For a graph $G$, we write $bk(G)$ for its maximum book size and $\tau(G)$ for the number of edges contained in triangles. Li, Liu and Zhang [J. Combin. Theory Ser. B 179 (2026) 219--249] proved that every $m$-edge Nosal graph satisfies $bk(G)> \frac{1}{24}\sqrt{m}$ and $\tau(G) > \frac{1}{12}\sqrt{m}$. Recently, two results on the booksize constant are proved: $\frac{1}{9}$ by Zhai, Li and Lou [arXiv:2601.10163v2], and $\frac{1}{4}$ by Chen, Li and Tang [arXiv:2607.16746v1]. In this paper, we establish the following result: Every $m$-edge graph $G$ with no isolated vertices and $\rho(G)\geq \sqrt{m}$ that is not isomorphic to any complete bipartite graph satisfies $bk(G)\geq\frac{\rho(G)}{3}$ and $\tau(G)\geq \rho(G)$. As direct consequences, we answer a question of Li, Liu and Zhang [J. Combin. Theory Ser. B 179 (2026) 219--249] and confirm a conjecture of Li, Feng and Peng [J. Graph Theory 110 (4) (2025) 408--425].

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Xinghui Zhao, Lihua You, Jing Zeng, Xiaoxue Zhang. 2026-07-16. Two problems on booksize and triangular edges in Nosal graphs. https://arxiv.org/abs/2607.15071

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