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arXiv 2008.07474math.CO
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关于横截临界图的一个定理的谱加强

Spectral strengthening of a theorem on transversal critical graphs

Muhuo Liu, Xiaofeng Gu

发表信息：
Discrete Mathematics 345 (2022), 112717
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摘要：

图$G$的横截集是与$G$的所有边关联的顶点的集合。$G$的横截数，记作$τ(G)$，是$G$的横截集的最小基数。一个没有孤立点的简单图$G$称为$τ$ -临界图，如果对$e\in E(G)$的每条边都有$τ(G-e) < τ(G)$.对于任何$τ$ -临界图$G$与$τ(G)=t$，它已被证明是$|V(G)|\le 2t$由Erdenses和Gallai和$|E(G)|\le {t+1\choose 2}$由Erdenses，Hajnal和月亮。最近，Gyárfás和Lehel将其扩展为$|V(G)| + |E(G)|\le {t+2\choose 2}$。在本文中，我们通过谱证明了更强的结果。令$G$是具有$τ(G)=t$和$|V(G)|=n$的$τ$ -临界图，并令$λ_1$表示$G$的邻接矩阵的最大特征值。我们证明了$n + λ_1\le 2t+1$与平等当且仅当$G$是$tK_2$，$K_{s+1}\cup (t-s)K_2$，或$C_{2s-1}\cup (t-s)K_2$，其中$2\leq s\leq t$ ;特别是，$λ_1(G)\le t$与平等当且仅当$G$是$K_{t+1}$。然后，我们应用它表明，对于任何非负整数$r$，我们有$n\left(r+ \frac{λ_1}{2}\right) \le {t+r+1\choose 2}$和所有的极值图的特征。这意味着一个纯组合结果$r|V(G)| + |E(G)| \le {t+r+1\choose 2}$，它比Erdens-Hajnal-Moon定理和Gyárfás-Lehel定理更强.我们还有其他一些概括。

英文摘要：

A transversal set of a graph $G$ is a set of vertices incident to all edges of $G$. The transversal number of $G$, denoted by $τ(G)$, is the minimum cardinality of a transversal set of $G$. A simple graph $G$ with no isolated vertex is called $τ$-critical if $τ(G-e) < τ(G)$ for every edge $e\in E(G)$. For any $τ$-critical graph $G$ with $τ(G)=t$, it has been shown that $|V(G)|\le 2t$ by Erdős and Gallai and that $|E(G)|\le {t+1\choose 2}$ by Erdős, Hajnal and Moon. Most recently, it was extended by Gyárfás and Lehel to $|V(G)| + |E(G)|\le {t+2\choose 2}$. In this paper, we prove stronger results via spectrum. Let $G$ be a $τ$-critical graph with $τ(G)=t$ and $|V(G)|=n$, and let $λ_1$ denote the largest eigenvalue of the adjacency matrix of $G$. We show that $n + λ_1\le 2t+1$ with equality if and only if $G$ is $tK_2$, $K_{s+1}\cup (t-s)K_2$, or $C_{2s-1}\cup (t-s)K_2$, where $2\leq s\leq t$; and in particular, $λ_1(G)\le t$ with equality if and only if $G$ is $K_{t+1}$. We then apply it to show that for any nonnegative integer $r$, we have $n\left(r+ \frac{λ_1}{2}\right) \le {t+r+1\choose 2}$ and characterize all extremal graphs. This implies a pure combinatorial result that $r|V(G)| + |E(G)| \le {t+r+1\choose 2}$, which is stronger than Erdős-Hajnal-Moon Theorem and Gyárfás-Lehel Theorem. We also have some other generalizations.

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