Super-edge-connectivity of G( k, d, s)(s ⩾ k/2)

Abstract

A graph G is super-edge-connected, for short super-λ, if every minimum edge-cut consists of edges adjacent to a vertex of minimum degree. Alphabet overlap graph G(k, d, s) is undirected, simple graph with vertex set \(V = \left\{ {\left. v \right|v = \left( {v_1 \cdots v_k } \right);v_i \in \left\{ {1,2, \cdots ,d} \right\},i = 1, \cdots ,k} \right\}\). Two vertices \(u = \left( {u_1 \cdots u_k } \right)\) and \(v = \left( {v_1 \cdots v_k } \right)\) are adjacent if and only if us+i =vi or \(v_{s + i} = u_i \left( {i = 1, \cdots ,k - s} \right)\). In particular G(k, d, 1) is just an undirected de Bruijn graph. In this paper, we show that the diameter of G(k, d, s) is \(\left\lceil {\frac{k} {s}} \right\rceil\), the girth is 3. Finally, we prove that G(k, d, s)(s ⩾ k /2) is super-λ.