On the excessive [m]-index of a tree

TL;DR

This paper investigates the excessive [m]-index of trees, providing a formula for m=4 and exploring the challenges of generalizing to other graphs, advancing understanding of edge covering by matchings.

Contribution

It introduces a formula for the excessive [4]-index of trees and discusses the limitations of extending this to arbitrary graphs, proposing a conjecture for general m.

Findings

A formula for the excessive [4]-index of trees is established.

The formula does not extend to all graphs, indicating complexity in general cases.

A conjecture for the excessive [m]-index of trees for any m is proposed.

Abstract

The excessive [m]-index of a graph G is the minimum number of matchings of size m needed to cover the edge-set of G. We call a graph G [m]-coverable if its excessive [m]-index is finite. Obviously the excessive [1]-index is |E(G)| for all graphs and it is an easy task the computation of the excessive [2]-index for a [2]-coverable graph. The case m=3 is completely solved by Cariolaro and Fu in 2009. In this paper we prove a general formula to compute the excessive [4]-index of a tree and we conjecture a possible generalization for any value of m. Furthermore, we prove that such a formula does not work for the excessive [4]-index of an arbitrary graph.