Strong and Weighted Matchings in Inhomogenous Random Graphs

TL;DR

This paper studies strong matchings in deterministic and inhomogeneous random graphs, providing bounds on their expected minimum weight and maximum size using probabilistic and combinatorial methods.

Contribution

It introduces bounds on strong matchings' weight and size in both deterministic and inhomogeneous random graphs, extending previous matching theory.

Findings

Bounds on the expectation and variance of minimum weight of maximum strong matchings.

Bounds on the maximum size of strong matchings in inhomogeneous random graphs.

Application of local neighborhoods, martingale differences, and exploration techniques.

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.