Uniquely bipancyclic graphs

Abstract

In this chapter we look at the problem of uniquely bipancyclic graphs, that is bipartite graphs that contain exactly one cycle of each length from 4 up to the number of vertices. If such a graph contains c cycles, their lengths are 4, 6, …, 2c + 2, so the graph has 2c + 2 vertices; moreover the sum of the lengths of the cycles (total number of edges in the cycles, with multiple appearances in different cycles counted multiply) is 4 + 6 + ⋯ + (2 c+ 2) = c2+ 3 c. We shall denote this by s(c).