On the expected number of edges in a maximum matching of an (r,s)-tree

Abstract

An (r, s)-tree is a connected, acyclic, bipartite graph with r light and s dark vertices. In an earlier paper [Pa92] we investigated the edge independence number of the superposition of 2 or more (r, s)-trees. Here we focus on just one (r, s)-tree. We use three variable exponential generating functions to establish a recurrence relation for the expected value of the vertex independence number of (r, s)-trees, and hence that of the edge independence number. Numerical values are calculated for small r and s. The data shows that for small (n, n)-trees the average percentage of dark vertices in a maximum matching is over 87%. This provides a basis for comparison with the asymptotic behavior of the expectation, a topic to be explored in another paper.