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Between proper and strong edge-colorings of subcubic graphs [Elektronski vir]
Hocquard, Herve ; Lajou, Dimitri ; Lužar, Borut
In a proper edge-coloring the edges of every color form a matching. A matching is induced if the endvertices 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 asserting that, for subcubic graphs, 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 eight induced matchings, and two matchings and at most five 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 abovementioned conjecture for class I graphs.
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