(R)-Pancyclic, (r)-Bipancyclic and Oddly (r)-Bipancyclic Graphs

摘要

A graph with v vertices is (r)-pancyclic if it contains precisely r cycles of every length from 3 to v. A bipartite graph with even number of vertices v is said to be (r)-bipancyclic if it contains precisely r cycles of each even length from 4 to v. A bipartite graph with odd number of vertices v and minimum degree at least 2 is said to be oddly (r)- bipancyclic if it contains precisely r cycles of each even length from 4 to v - 1. In this paper, using a computer search, we classify all (r)-pancyclic and (r)-bipancyclic graphs, r ≥ 2, with v vertices and at most v+5 edges. We also classify all oddly (r)-bipancyclic graphs, r ≥ 1, with v vertices and at most v + 4 edges.