Excessive [l,m]-factorizations

TL;DR

This paper introduces the excessive [l,m]-index, a new graph parameter measuring the minimal number of matchings needed to cover a graph with matchings of size between l and m, and provides a formula and polynomial algorithm for it.

Contribution

It defines the excessive [l,m]-index, derives a general formula for it, and presents a polynomial-time algorithm to compute and construct the corresponding factorization.

Findings

Provides a general formula for the excessive [l,m]-index.

Develops a polynomial-time algorithm for computing the index.

Outputs an excessive [l,m]-factorization when it exists.

Abstract

Given two positive integers l and m, with l \le m, an [l,m]-covering of a graph G is a set M of matchings of G whose union is the edge set of G and such that l \le |L| \le m for every matching L of M. An [l,m]-covering M of G is an excessive [l,m]-factorization of G if the cardinality of M is as small as possible. The number of matchings in an excessive [l,m]-factorization of G (or \infty, if G does not admit an excessive [l,m]-factorization) is a graph parameter called the excessive [l,m]-index of G and denoted by \chi'[l,m](G). In this paper we study such parameter. Our main result is a general formula for the excessive [l,m]-index of a graph G in terms of other graph parameters. Furthermore, we give a polynomial time algorithm which computes \chi'[l,m](G) and outputs an excessive [l,m]-factorization of G, whenever the latter exists.