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- W3049219688 abstract "A transversal set of a graph $G$ is a set of vertices meeting all edges of $G$. The transversal number of $G$, denoted by $tau(G)$, is the minimum cardinality of a transversal set of $G$. A simple graph $G$ with no isolated vertex is called $tau$-critical if $tau(G-e) < tau(G)$ for every edge $ein E(G)$. For any $tau$-critical graph $G$ with $tau(G)=t$, it has been shown that $|V(G)|le 2t$ by Erdős and Gallai and that $|E(G)|le {t+1choose 2}$ by Erdős, Hajnal and Moon. Most recently, it was improved by Gyarfas and Lehel with a very short proof that $|V(G)| + |E(G)|le {t+2choose 2}$ [J. Graph Theory, this https URL]. They also determined all extremal graphs. In this paper, we improve these results via spectrum and determine all extremal graphs. Let $lambda_1$ denote the largest eigenvalue of the adjacency matrix of $G$. We show that for any $tau$-critical graph $G$ with $tau(G)=t$ and $|V(G)|=n$, we have $n + lambda_1le 2t+1$ with equality holds 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 $2leq sleq t$. We then apply it to show that for any nonnegative integer $r$ and any $tau$-critical graph $G$ with $tau(G)=tge r$ and $|V(G)|=n$, we have $nleft(r+ frac{lambda_1}{2}right) le {t+r+1choose 2}$ and characterize all extremal graphs. This result implies Erdős-Hajnal-Moon Theorem and Gyarfas-Lehel Theorem as well as a more general result that $r|V(G)| + |E(G)| le {t+r+1choose 2}$ with the extremal graphs characterized. We also have some other generalizations." @default.
- W3049219688 created "2020-08-21" @default.
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- W3049219688 date "2020-08-17" @default.
- W3049219688 modified "2023-09-27" @default.
- W3049219688 title "Spectral strengthening of a theorem on transversal critical graphs" @default.
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