A new proof of Melnikovʹs conjecture on the edge-face coloring of plane graphs Original Research Article

In 1975, Melnikov conjectured that the edges and faces of each plane graph G may be colored with Δ(G)+3 colors so that any two adjacent or incident elements receive different colors, where Δ(G) is the maximum degree of G. Two similar, yet independent, proofs of this conjecture have been published recently by Waller (J. Combin. Theory Ser. B 69 (1997) 219) and Sanders and Zhao (Combinatorica 17 (1997) 441). Both proofs made use of the Four-Color Theorem. This paper presents a new proof of Melnikovʹs conjecture independent of the Four-Color Theorem.