Some sufficient conditions for a planar graph of maximum degree six to be Class 1

Let G be a planar graph of maximum degree 6. In this paper we prove that if G   does not contain either a 6-cycle, or a 4-cycle with a chord, or a 5- and 6-cycle with a chord, then χ′(G)=6χ′(G)=6, where χ′(G)χ′(G) denotes the chromatic index of G.