The Chromatic Index of a Claw-Free Graph Whose Core has Maximum Degree $$2$$ 2

Abstract

Let $$G$$G be a graph. The core of $$G$$G, denoted by $$G_{\Delta }$$GΔ, is the subgraph of $$G$$G induced by the vertices of degree $$\Delta (G)$$Δ(G), where $$\Delta (G)$$Δ(G) denotes the maximum degree of $$G$$G. A $$k$$k-edge coloring of $$G$$G is a function $$f:E(G)\rightarrow L$$f:E(G)?L such that $$|L| = k$$|L|=k and $$f(e_1)\ne f(e_2)$$f(e1)?f(e2), for any two adjacent edges $$e_1$$e1 and $$e_2$$e2 of $$G$$G. The chromatic index of $$G$$G, denoted by $$\chi '(G)$$??(G), is the minimum number $$k$$k for which $$G$$G has a $$k$$k-edge coloring. A graph $$G$$G is said to be Class $$1$$1 if $$\chi '(G) = \Delta (G)$$??(G)=Δ(G) and Class $$2$$2 if $$\chi '(G) = \Delta (G) + 1$$??(G)=Δ(G)+1. Hilton and Zhao conjectured that if $$G$$G is a connected graph, $$\Delta (G_{\Delta })\le 2$$Δ(GΔ)≤2, and $$G$$G is not the Petersen graph with one vertex removed, then $$G$$G is Class $$2$$2 if and only if $$G$$G is overfull. In this paper, we prove this conjecture for claw-free graphs of even order.