不含2-因子且坚韧度趋近于2的无三角形图
Triangle-Free Graphs of Toughness Approaching Two Without a 2-Factor
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中文总结 AI 辅助
本文证实了Bauer等人1996年的猜想，构造出坚韧度趋近于2且不含2-因子的无三角形图，说明存在2-因子的阈值2在无三角形图类中仍为最优界限。
中文摘要 AI 辅助
根据Enomoto、Jackson、Katerinis与Saito在1985年的研究，每一个2-坚韧图都存在2-因子，且该坚韧度界限是最优的：对任意ε>0，都存在(2-ε)-坚韧图不含2-因子。自然会问该结论对无三角形图是否仍成立，Bauer、van den Heuvel与Schmeichel在1996年提出此猜想，同时给出一个无三角形图的无限族，认为其坚韧度趋近于2，但未确立所需的坚韧度界限。本文证实了该猜想：对每个偶数q≥6，构造了不含2-因子的无三角形图G_q，其坚韧度τ(G_q)=(2q²-q-2)/(q²+q)=2-(3q+2)/(q²+q)。特别地，当q→∞时，τ(G_q)→2，表明存在2-因子的阈值2在无三角形图类中仍是最优的。
英文摘要
By work of Enomoto, Jackson, Katerinis, and Saito from 1985, every $2$-tough graph has a $2$-factor, and this toughness bound is best possible: for every $\varepsilon>0$, there exist $(2-\varepsilon)$-tough graphs with no $2$-factor. It is natural to ask whether the latter statement remains true for triangle-free graphs. Bauer, van den Heuvel, and Schmeichel conjectured this in 1996. In the same paper, they proposed an infinite family of triangle-free graphs with no $2$-factor whose toughness they believed approaches $2$, but the required toughness bound was not established. In this paper, we confirm their conjecture. For every even integer $q\ge 6$, we construct a triangle-free graph $G_q$ with no $2$-factor and with toughness \[ τ(G_q) =\frac{2q^2-q-2}{q^2+q} =2-\frac{3q+2}{q^2+q}. \] In particular, $τ(G_q)\to 2$ as $q\to\infty$, showing that the threshold $2$ for the existence of a $2$-factor remains best possible even within the class of triangle-free graphs.
