Some criteria for a graph to be Class 1

Let GG be a graph. The core of GG, denoted by GΔGΔ, is the subgraph of GG induced by the vertices of degree Δ(G)Δ(G), where Δ(G)Δ(G) is the maximum degree of GG. A kk-edge coloring   of a graph GG is a function f:E(G)⟶Lf:E(G)⟶L, where ∣L∣=k∣L∣=k and f(e1)≠f(e2)f(e1)≠f(e2), for every two adjacent edges e1,e2e1,e2 of GG. The edge chromatic number of  GG, denoted by χ′(G)χ′(G), is the minimum number kk for which GG has a kk-edge coloring. A graph GG is said to be Class  11 if χ′(G)=Δ(G)χ′(G)=Δ(G) and Class  22 if χ′(G)=Δ(G)+1χ′(G)=Δ(G)+1. In this paper, it is shown that, for every connected graph of even order, if GΔ=C6GΔ=C6, then GG is Class 11. Also, we prove that, if GG is a connected graph, and every connected component of GΔGΔ is a unicyclic graph or a tree, and GΔGΔ is not a disjoint union of cycles, then GG is Class 11.