AI 中文总结
本文研究Nosal图中书尺寸和三角形边的问题，证明了无孤立顶点且非完全二分图的Nosal图的书尺寸和三角形边数下限。
AI 中文摘要
一个具有m条边的图G被称为Nosal图，如果ρ(G) > √m。对于一个图G，我们用bk(G)表示其最大书尺寸，用τ(G)表示包含在三角形中的边数。Li、Liu和Zhang[J. Combin. Theory Ser. B 179 (2026) 219--249]证明了每一个m边的Nosal图满足bk(G) > (1/24)√m和τ(G) > (1/12)√m。最近，Zhai、Li和Lou[arXiv:2601.10163v2]证明了每一个m边的Nosal图满足bk(G) > (1/9)√m。在本文中，我们得出以下结果：每一个没有孤立顶点且ρ(G) ≥ √m且不与任何完全二分图同构的m边图G满足bk(G) ≥ ρ(G)/3和τ(G) ≥ ρ(G)。作为直接结果，我们回答了Li、Liu和Zhang[J. Combin. Theory Ser. B 179 (2026) 219--249]提出的问题，并确认了Li、Feng和Peng[J. Graph Theory 110 (4) (2025) 408--425]的猜想。
英文摘要
A graph $G$ with $m$ edges is said to be a Nosal graph if $ρ(G)>\sqrt{m}$. For a graph $G$, we write $bk(G)$ for its maximum book size and $τ(G)$ 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 $τ(G) > \frac{1}{12}\sqrt{m}$. 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 $ρ(G)\geq \sqrt{m}$ that is not isomorphic to any complete bipartite graph satisfies $bk(G)\geq\frac{ρ(G)}{3}$ and $τ(G)\geq ρ(G)$. 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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