Spectral strengthening of a theorem on transversal critical graphs
Abstract
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∈ E(G). For any τ-critical graph G with τ(G)=t, it has been shown that |V(G)| 2t by Erdos and Gallai and that |E(G)| t+1 2 by Erdos, Hajnal and Moon. Most recently, it was extended by Gy\'arf\'as and Lehel to |V(G)| + |E(G)| t+2 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 λ1 denote the largest eigenvalue of the adjacency matrix of G. We show that n + λ1 2t+1 with equality if and only if G is tK2, Ks+1 (t-s)K2, or C2s-1 (t-s)K2, where 2≤ s≤ t; and in particular, λ1(G) t with equality if and only if G is Kt+1. We then apply it to show that for any nonnegative integer r, we have n(r+ λ12) t+r+1 2 and characterize all extremal graphs. This implies a pure combinatorial result that r|V(G)| + |E(G)| t+r+1 2, which is stronger than Erdos-Hajnal-Moon Theorem and Gy\'arf\'as-Lehel Theorem. We also have some other generalizations.