Cycles Containing Three Consecutive Edges in 2k-Edge-Connected Graphs

Abstract

We consider finite undirected graphs possibly with multiple edges but without loops. Let G be a graph and let V=V(G) and E=E(G) be the set of vertices and edges of G respectively. We allow repetition of vertices (but not edges) in a path and cycle. λ(G) denotes the edge-connectivity of G. For X, Y⊂ V(G), with X ∩ Y= o, ∂(X,Y) denotes the set of edges between X and Y. ∂(X) denotes ∂(X,V(G)−X), and is called a cut. A cut ∂(X) is called an n-cut if |∂(X)|=n, and will be called nontrivial if |X|≥ 2 and |V(G)−X| ≥2. We set e(X,Y):= |∂ (X,Y)| and e(X)= |∂(X)|. For an f ∈ E(G), V(f) denotes the set of end vertices of f. Cycles containing two (adjacent) edges in 2k-(k-)edgeconnected graphs reducing the edge-connectivity at most two are investigated in [3] ([2] and Mader [1]). We here consider cycles containing three consecutive edges.