Between proper and strong edge‐colorings of subcubic graphs

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 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.

Every subcubic multigraph is (1,27) $(1,{2}^{7})$‐packing edge‐colorable

“…In addition, Gastineau and Togni in [13] showed that every subcubic graph $$G$$ with a 2‐factor has a $$({1}^{2},{2}^{5})$$‐packing edge‐coloring, and if the graph is additionally 3‐edge‐colorable, then it is $$({1}^{2},{2}^{4})$$‐packing edge‐colorable. Hocquard et al [14, 15] were able to show the same holds without the assumption of a 2‐factor. They also restate the following conjectures of Gastineau and Togni.…”

“…2 5 -packing edge-coloring, and if the graph is additionally 3-edge-colorable, then it is (1 , 2 ) 2 4 -packing edge-colorable. Hocquard et al [14,15] were able to show the same holds without the assumption of a 2-factor. They also restate the following conjectures of Gastineau and Togni.…”

“…There are several current conjectures regarding $$S$$‐packing edge‐colorings for subcubic graphs. For example, Hocquard et al in [15] show that every 3‐edge‐colorable subcubic graph is $$(1,{2}^{7})$$‐packing edge‐colorable, and pose the following conjecture.…”

“…An immediate corollary of the result “every subcubic graph has a $$({2}^{10})$$‐packing edge‐coloring” is that every subcubic graph has a $$(1,{2}^{9})$$‐packing edge‐coloring. Recently, Hocquard, Lajou, and Lužar in [15] showed that every subcubic graph has a $$(1,{2}^{8})$$‐packing edge‐coloring, and posed the following conjecture, which was initially posed as a question by Gastineau and Togni in [13]:…”

“…Conjecture 28 (Hocquard, Lajou, and Lužar [15]). Every 3-edge-colorable subcubic graph is (1, 2 ) 6 -packing edge colorable.…”