Covering cubic graphs with matchings of large size

TL;DR

This paper investigates the minimum number of large matchings needed to cover all edges of cubic graphs, focusing on the case where matchings are of size n-1, and explores its relation to graph connectivity and other parameters.

Contribution

It studies the excessive [n-1]-index in cubic graphs, providing bounds for low connectivity graphs and conjecturing limits for cyclically 4-connected graphs.

Findings

Excessive [n-1]-index can be large in low connectivity cubic graphs.

Cyclically 4-connected cubic graphs likely have excessive [n-1]-index at most 4.

Relation established between excessive [n-1]-index, oddness, and circumference.

Abstract

Let m be a positive integer and let G be a cubic graph of order 2n. We consider the problem of covering the edge-set of G with the minimum number of matchings of size m. This number is called excessive [m]-index of G in literature. The case m=n, that is a covering with perfect matchings, is known to be strictly related to an outstanding conjecture of Berge and Fulkerson. In this paper we study in some details the case m=n-1. We show how this parameter can be large for cubic graphs with low connectivity and we furnish some evidence that each cyclically 4-connected cubic graph of order 2n has excessive [n-1]-index at most 4. Finally, we discuss the relation between excessive [n-1]-index and some other graph parameters as oddness and circumference.