On vertex types and cyclic colourings of 3-connected plane graphs Original Research Article

We prove a structural theorem of Lebesgueʹs type concerning the existence of certain types of vertices in 3-connected plane graphs. This theorem is then applied to the proof such that the cyclic chromatic number χc(G)⩽k+2 for k=max{40,Δ∗(G)}, where Δ∗(G) denotes the size of the largest face of G. This confirms a conjecture by Plummer and Toft for Δ∗(G)⩾40.