Graphs and Combinatorics

, Volume 22, Issue 3, pp 289–295

# Coloring Vertices and Faces of Locally Planar Graphs

• Michael O. Albertson
• Bojan Mohar
Article

## Abstract

If G is an embedded graph, a vertex-face r-coloring is a mapping that assigns a color from the set {1, . . . ,r} to every vertex and every face of G such that different colors are assigned whenever two elements are either adjacent or incident. Let χ vf (G) denote the minimum r such that G has a vertex-face r-coloring. Ringel conjectured that if G is planar, then χ vf (G)≤6. A graph G drawn on a surface S is said to be 1-embedded in S if every edge crosses at most one other edge. Borodin proved that if G is 1-embedded in the plane, then χ(G)≤6. This result implies Ringel's conjecture. Ringel also stated a Heawood style theorem for 1-embedded graphs. We prove a slight strengthening of this result. If G is 1-embedded in S, let w(G) denote the edge-width of G, i.e. the length of a shortest non-contractible cycle in G. We show that if G is 1-embedded in S and w(G) is large enough, then the list chromatic number ch(G) is at most 8.

## Keywords

Vertex face coloring 1-embedded Locally planar

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