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

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 an 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 an 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, \(r \geq 2\), with \(v\) vertices and at most \(v+5\) edges. We also classify all oddly \((r)\)-bipancyclic graphs, \(r \geq 1\), with \(v\) vertices and at most \(v+4\) edges.