Strongly maximal matchings in infinite weighted graphs

TL;DR

This paper proves that in infinite weighted graphs with finitely many rational weights, there exists a matching that is strongly maximal, meaning no other matching can improve the weight difference in a specific way.

Contribution

It establishes the existence of strongly w-maximal matchings in infinite graphs under rational weight conditions, a novel extension in graph theory.

Findings

Existence of strongly w-maximal matchings in infinite graphs with rational weights

The result applies to graphs with finitely many weight values

Provides a new approach to maximal matchings in infinite settings

Abstract

Given an assignment of weights w to the edges of a graph G, a matching M in G is called strongly w-maximal if for any matching N the sum of weights of the edges in N\M is at most the sum of weights of the edges in M\N. We prove that if w assumes only finitely many values all of which are rational then G has a strongly w-maximal matching.