The Chromatic Index of a Graph Whose Core is a Cycle of Order at Most 13

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) 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 (e ₁) ≠ f (e ₂) for all two adjacent edges e ₁ and e ₂ 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) = Δ(G) and Class 2 if χ′(G) = Δ(G) + 1. In this paper it is shown that every connected graph G of even order whose core is a cycle of order at most 13 is Class 1.