The chromatic index of a graph whose core has maximum degree two

Let G be a graph. The core of G, denoted by G∆, is the subgraph of G induced by the vertices of degree ∆(G), where ∆(G) denotes the maximum degree of G. A k-edge coloring of G is a function f : E(G)→ L such that |L| = k and f(e1) 6= f(e2) for all two adjacent edges e1 and e2 of G. The chromatic index 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) = ∆(G) and Class 2 if χ′(G) = ∆(G)+1. In this paper it is shown that every connected graph G of even order and with ∆(G∆) ≤ 2 is Class 1 if |G∆| ≤ 9 or G∆ is a cycle of order 10.