Some criteria for a graph to be Class 1

Abstract

Let G be a graph. The core of G, denoted by GΔ, is the subgraph of G induced by the vertices of degree Δ(G), where Δ(G) is the maximum degree of G. A k-edge coloring of a graph G is a function f:E(G)⟶L, where ∣L∣=k and f(e1)≠f(e2), for every two adjacent edges e1,e2 of G. The edge chromatic number ofG, denoted by χ′(G), is the minimum number k for which G has a k-edge coloring. A graph G is said to be Class1 if χ′(G)=Δ(G) and Class2 if χ′(G)=Δ(G)+1. In this paper, it is shown that, for every connected graph of even order, if GΔ=C6, then G is Class 1. Also, we prove that, if G is a connected graph, and every connected component of GΔ is a unicyclic graph or a tree, and GΔ is not a disjoint union of cycles, then G is Class 1.