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.
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Work completed while the author was the Neil R. Grabois Visiting Chair of Mathematics, Colgate University, Hamilton, NY 13346 USA.
Supported in part by the Ministry of Science and Higher Education of Slovenia, Research Program P1–0507–0101.
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Albertson, M., Mohar, B. Coloring Vertices and Faces of Locally Planar Graphs. Graphs and Combinatorics 22, 289–295 (2006). https://doi.org/10.1007/s00373-006-0653-4
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DOI: https://doi.org/10.1007/s00373-006-0653-4