Simultaneously Colouring the Edges and Faces of Plane Graphs

The elements of a plane graph G are the edges, vertices, and faces of G. We say that two elements are neighbours in G if they are incident with or are mutually adjacent with each other in G. The simultaneous colouring of distinct elements of a planar graph was first introduced by Ringel [12]. In his paper Ringel considered the problem of colouring the vertices and faces of a plane graph in such a way that every vertex and face receives a different colour from any of its neighbouring vertices and faces. Ringels conjecture that six colours always suffice for this colouring was proved by Borodin [2], this bound being best possible. In this paper we consider the simultaneous colouring of edges and faces of a plane graph, where every edge and face receives a different colour from any of its neighbouring edges and faces. We define /ef (G) to be the least number of colours needed for such a colouring of the plane graph G. This problem was first studied by Jucovic [8] and Fiamcik [7] for 3and 4-regular graphs. Mel'nikov [11] conjectured that /ef (G) 2+3 for every plane graph G, where 2 is the maximum degree of a vertex in G. It is easy to see that this bound is attained by odd cycles. Borodin [3] proved this conjecture for the special case 2 3 (this result was also proved independently by Lin et al. [9]). In [4, 5] Borodin gave further results on this article no. TB971725