Super \(3\)-Restricted Edge Connectivity of Triangle-Free Graphs

Abstract

Let \(G = (V, E)\) be a connected graph. An edge set \(S \subset E\) is a \(k\)-restricted edge cut if \(G – S\) is disconnected and every component of \(G – S\) has at least \(k\) vertices. The \(k\)-restricted edge connectivity \(\lambda_k(G)\) of \(G\) is the cardinality of a minimum \(k\)-restricted edge cut of \(G\). A graph \(G\) is \(\lambda_k\)-connected if \(k\)-restricted edge cuts exist. A graph \(G\) is called \(\lambda_k\)-optimal if \(\lambda_k(G) = \xi_k(G)\), where \[\xi_k(G) = \min\{|[X, Y]|: X \subseteq V, |X| = k \text{ and } G[X] \text{ is connected}\};\] Here, \(G[X]\) is the subgraph of \(G$\) induced by the vertex subset \(X \subseteq V\), and \(Y = V \setminus X\) is the complement of \(X\); \([X, Y]\) is the set of edges with one end in \(X\) and the other in \(Y\). \(G\) is said to be super-\(\lambda_k\) if each minimum \(k\)-restricted edge cut isolates a connected subgraph of order \(k\). In this paper, we give some sufficient conditions for triangle-free graphs to be super-\(\lambda_3\).