Some criteria for a graph to be Class 1

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 of G, denoted by ^χ′(G), is the minimum number k for which G has a k-edge coloring. A graph G is said to be Class 1 if ^χ′(G)= Δ(G) and Class 2 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.