A New Proof for a Result on the Inclusion Chromatic Index of Subcubic Graphs

Let G be a graph with a minimum degree δ of at least two. The inclusion chromatic index of G, denoted by χ⊂′(G), is the minimum number of colors needed to properly color the edges of G so that the set of colors incident with any vertex is not contained in the set of colors incident to any of its neighbors. We prove that every connected subcubic graph G with δ(G)≥2 either has an inclusion chromatic index of at most six, or G is isomorphic to K^2,3, where its inclusion chromatic index is seven.