Between proper and strong edge-colorings of subcubic graphs

TL;DR

This paper explores intermediate edge-colorings in subcubic graphs, proving bounds on decompositions into matchings and induced matchings, and confirming a conjecture for class I graphs.

Contribution

It introduces new bounds on decompositions of subcubic graphs into matchings and induced matchings, and verifies a conjecture for class I graphs.

Findings

Every maximum degree 3 graph decomposes into one matching and at most 8 induced matchings.

Such graphs can also be decomposed into two matchings and at most 5 induced matchings.

The conjecture holds for class I graphs, reducing the number of induced matchings needed.

Abstract

In a proper edge-coloring the edges of every color form a matching. A matching is induced if the end-vertices of its edges induce a matching. A strong edge-coloring is an edge-coloring in which the edges of every color form an induced matching. We consider intermediate types of edge-colorings, where edges of some colors are allowed to form matchings, and the remaining form induced matchings. Our research is motivated by the conjecture proposed in a recent paper of Gastineau and Togni on S-packing edge-colorings (On S-packing edge-colorings of cubic graphs, Discrete Appl. Math. 259 (2019), 63-75) asserting that by allowing three additional induced matchings, one is able to save one matching color. We prove that every graph with maximum degree 3 can be decomposed into one matching and at most 8 induced matchings, and two matchings and at most 5 induced matchings. We also show that if a graph is in class I, the number of induced matchings can be decreased by one, hence confirming the above-mentioned conjecture for class I graphs.