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.
Highlights
Matchings in graphs is an important object of study from both theoretical and application perspectives
We equip the edges of a deterministic graph H with independent but not necessarily identically distributed weights and study a generalized version of matchings in H satisfying the property that end-vertices of any two distinct edges are at least a minimum distance apart
In the first part of our paper, we study minimum weight of a strong matchings where end-vertices of distinct edges are at a given minimum distance apart
Summary
Matchings in graphs is an important object of study from both theoretical and application perspectives. One of the most well-studied aspect of matchings from the probabilistic perspective is that of minimum weight matchings [1]: Given a complete bipartite graph on n + n vertices and equipping each edge with an independent exponential weight, the problem is to determine the minimum weight C(n) of a perfect matching It is well known [10, 11] that EC(n) =. [6] used a combination of second moment method along with concentration inequalities to estimate the largest possible size of induced matchings in homogenous graphs and obtained deviation bounds for a wide range of edge probabilities.
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