Enumeration of the Bases of the Bicircular Matroid on a Complete Bipartite Graph

Abstract

We enumerate the bases of the bicircular matroid on \(K_{m,n}\). The structure of bases of the bicircular matroid in relation to the bases of the cycle matroid is explored. The techniques herein may enable the enumeration of the bases of bicircular matroids on larger classes of graphs; indeed one of the motivations for this work is to show the extendibility of the techniques recently used to enumerate the bases of the bicircular matroid on \(K_n\).