# Coloring Vertices and Faces of Locally Planar Graphs

- Received:
- Accepted:

DOI: 10.1007/s00373-006-0653-4

- Cite this article as:
- Albertson, M. & Mohar, B. Graphs and Combinatorics (2006) 22: 289. doi:10.1007/s00373-006-0653-4

## 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.