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

Lily Chen,Yanyi Li

doi:10.3390/axioms11010033

Abstract

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.

Highlights

Let G be a counterexample with a minimal number of edges, by establishing a series of auxiliary claims, they showed that G does not contain a 2-vertex adjacent to two 2-vertices, and any 3-vertex of G cannot be adjacent to a
We prove the result by contradiction
We show that G does not contain a 2-vertex adjacent to two 2-vertices, i.e., G contains no k-thread with k ≥ 3, and G does not contain a 3-cycle with one 2-vertex, and a 4-cycle with two non-adjacent 2-vertices

Summary

Introduction

Let G be a graph with minimum degree δ ≥ 2 and let φ be a proper edge coloring of G. The minimum number of colors required in an inclusion-free edge coloring of G is called the inclusion chromatic index and is denoted by χ0⊂ ( G ). The AVD-edge coloring has attracted the attention of several groups of graph theorists It was conjectured by Zhang et al [12] that χ0a ( G ) ≤ ∆ + 2 for any connected graph G with. Let G be a counterexample with a minimal number of edges, by establishing a series of auxiliary claims, they showed that G does not contain a 2-vertex adjacent to two 2-vertices, and any 3-vertex of G cannot be adjacent to a.

Proof of the Main Result

Conclusions