Some criteria for a graph to be Class 1

Let G be a graph. The core of G , denoted by G Δ , 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 ( e 1 ) ≠ f ( e 2 ) , for every two adjacent edges e 1 , e 2 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 G Δ = C 6 , then G is Class 1 . Also, we prove that, if G is a connected graph, and every connected component of G Δ is a unicyclic graph or a tree, and G Δ is not a disjoint union of cycles, then G is Class 1 .