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Coloring Vertices and Faces of Locally Planar Graphs

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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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References

  1. Albertson, M.O., Hutchinson, J.P.: Extending precolorings of subgraphs of locally planar graphs. Europ J Combinatorics 25, 863–871 (2004)

    Google Scholar 

  2. Albertson, M.O., Hutchinson, J.P.: The independence ratio and genus of a graph. Trans. Amer. Math. Soc. 226, 161–173 (1977)

    Google Scholar 

  3. Böhme, T., Mohar, B., Stiebitz, M.: Dirac's map-color theorem for choosability. J. Graph Theory 32, 327–339 (1999)

    Google Scholar 

  4. Borodin, O.: A new proof of a 6-color theorem. J. Graph Theory 19, 507–521 (1995)

    Google Scholar 

  5. Borodin, O., Kostochka, A., Raspaud, A., Sopena, E.: Acyclic colouring of 1-planar graphs. Disc Appl Math 114, 29–41 (2001)

    Google Scholar 

  6. Hutchinson, J.: personal communication

  7. Jensen, T.R., Toft, B.: Graph Coloring Problems. Wiley, New York, 1995

  8. Korzhik, V.: An infinite series of surfaces with known 1-chromatic number. J. Combin. Theory Ser. B bf 72, 80–90 (1998)

    Google Scholar 

  9. Korzhik, V.: On the 1-chromatic number of nonorientable surfaces with large genus, manuscript

  10. Kostochka, A.V., Stiebitz, M.: A new lower bound on the number of edges in colour-critical graphs and hypergraphs. J. Combin Theory Ser. B 87, 374–402 (2003)

    Google Scholar 

  11. Kostochka, A.V., Stiebitz, M.: A list version of Dirac's theorem on the number of edges in colour-critical graphs. J. Graph Theory 39, 165–177 (2002)

    Google Scholar 

  12. Lam, P.C.B., Zhang, Z.: The vertex-face total chromatic number of Halin graphs. Networks 30, Ê167–170 (1997)

    Google Scholar 

  13. Mohar, B., Thomassen, C.: Graphs on Surfaces. Johns Hopkins Univ Press, Baltimore, 2001

  14. Ringel, G.: A six-colour problem on the sphere, Theory of Graphs (Proc. Colloq. Tihany, 1966). Academic Press, New York, pp 265–269

  15. Ringel, G.: A nine color theorem for the torus and Klein bottle, The Theory and Applications of Graphs (Kalamazoo, Mich. 1980). Wiley, New York, 1981, pp 507–515

  16. Schumacher, H.: About the chromatic number of 1-embeddable graphs, Graph theory in memory of G.A. Dirac (Sandbjerg, 1985), 397–400, Ann. Discrete Math., 41, North-Holland, Amsterdam, 1989

  17. Schumacher, H.: Ein 7-Farbenzatz 1-einbettbarer Graphen auf der projektiven Ebene. Abh. Math. Sem. Univ. Hamburg 54, 5–14 (1984)

    Google Scholar 

  18. Thomassen, C.: Five-coloring maps on surfaces. J. Combin. Theory Ser. B 59, 89–105 (1993)

    Google Scholar 

  19. Wang, W., Liu, J.: On the vertex face total chromatic number of planar graphs. J. Graph Theory 22, 29–37 (1996)

    Google Scholar 

Download references

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Correspondence to Michael O. Albertson.

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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

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