Formula omitted]-restricted edge connectivity in [formula omitted]-clique-free graphs

Abstract

Let G be a graph with vertex set V(G) and edge set E(G). An edge subset S⊆E(G) is called a k-restricted edge cut if G−S is not connected and every component of G−S has at least k vertices. The k-restricted edge connectivity of a connected graph G, denoted by λk(G), is defined as the cardinality of a minimum k-restricted edge cut. Let [X,X̄] denote the set of edges between a vertex set X⊂V(G) and its complement X̄=V(G)∖X. A vertex set X⊂V(G) is called a λk-fragment if [X,X̄] is a minimum k-restricted edge cut of G. Let ξk(G)=min{|[X,X̄]|:|X|=k,G[X]is connected}. In this work, we give a lower bound on the cardinality of λk-fragments of a graph G satisfying λk(G)<ξk(G) and containing no (p+1)-cliques. As a consequence of this result, we show a sufficient condition for a graph G with λk(G)=ξk(G).