Bipancyclic graphs

Abstract

Recall that a bipartite graph is a graph with two disjoint sets of vertices, V1 and V2 say, where no two vertices in the same set are adjacent, and is a balanced bipartite graph if V1 and V2 are equal in size. Obviously a bipartite graph cannot be pancyclic, as it can contain no odd cycles. So, in 1982, Schmeichel and Mitchem (J Graph Theory 6, 429–439) defined a bipancyclic graph G to be a balanced bipartite graph that contains cycles of all even orders from 4 up to and including the number of vertices of G. A minimally bipancyclic graph is one with the smallest possible number of edges, given its number of vertices, and a uniquely bipancyclic graph is one with exactly one cycle of every possible order. The complete bipartite graph Kn, n is obviously bipancyclic when n ≥ 2.