Strong and Weighted Matchings in Inhomogenous Random Graphs

Abstract

We equip the edges of a deterministic graph $H$ with independent but not necessarily identically distributed weights and study a generalized version of matchings (i.e. a set of vertex disjoint edges) in $H$ satisfying the property that end-vertices of any two distinct edges are at least a minimum distance apart. We call such matchings as strong matchings and determine bounds on the expectation and variance of the minimum weight of a maximum strong matching. Next, we consider an inhomogenous random graph whose edge probabilities are not necessarily the same and determine bounds on the maximum size of a strong matching in terms of the averaged edge probability. We use local vertex neighbourhoods, the martingale difference method and iterative exploration techniques to obtain our desired estimates.