Advertisement

Graphs and Combinatorics

, Volume 35, Issue 3, pp 677–688 | Cite as

A Structure of 1-Planar Graph and Its Applications to Coloring Problems

  • Xin ZhangEmail author
  • Bei Niu
  • Jiguo Yu
Original Paper
  • 57 Downloads

Abstract

A graph is 1-planar if it can be drawn on a plane so that each edge is crossed by at most one other edge. In this paper, we first give a useful structural theorem for 1-planar graphs, and then apply it to the list edge and list total coloring, the (p, 1)-total labelling, and the equitable edge coloring of 1-planar graphs. More precisely, we verify the well-known List Edge Coloring Conjecture and List Total Coloring Conjecture for 1-planar graph with maximum degree at least 18, prove that the (p, 1)-total labelling number of every 1-planar graph G is at most \(\Delta (G)+2p-2\) provided that \(\Delta (G)\ge 8p+2\) and \(p\ge 2\), and show that every 1-planar graph has an equitable edge coloring with k colors for any integer \(k\ge 18\). These three results respectively generalize the main theorems of three different previously published papers.

Keywords

1-Planar graph List edge coloring List total coloring (p, 1)-Total labelling Equitable edge coloring 

Mathematics Subject Classification

05C15 05C10 

Notes

References

  1. 1.
    Bazzaro, F., Montassier, M., Raspaud, A.: \((d,1)\)-Total labelling of planar graphs with large girth and high maximum degree. Discrete Math. 307, 2140–2151 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bondy, J.A., Murty, U.S.R.: Graph Theory with Applications. North-Holland, New York (1976)CrossRefzbMATHGoogle Scholar
  3. 3.
    Borodin, O.V.: A new proof of the 6 color theorem. J. Graph Theory 19(4), 507–521 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Borodin, O.V.: Solution of Ringel’s problems on the vertex-face coloring of plane graphs and on the coloring of \(1\)-planar graphs. Diskret. Analiz 41, 12–26 (1984). (in Russian)zbMATHGoogle Scholar
  5. 5.
    Borodin, O.V., Kostochka, A.V., Woodall, D.R.: List edge and list total colourings of multigraphs. J. Combin. Theory Ser. B 71, 184–204 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Fabrici, I., Madaras, T.: The structure of 1-planar graphs. Discrete Math. 307, 854–865 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Havet, F.: \((d,1)\)-total labelling of graphs. In: Workshop on Graphs and Algorithm, Dijon (2003)Google Scholar
  8. 8.
    Havet, F., Yu, M.-L.: \((d,1)\)-total labelling of graphs, Technical Report 4650, INRIA (2002)Google Scholar
  9. 9.
    Havet, F., Yu, M.-L.: \((p,1)\)-total labelling of graphs. Discrete Math. 308, 496–513 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Hu, D.-Q., Wu, J.-L., Yang, D., Zhang, X.: On the equitable edge-coloring of 1-planar graphs and planar graphs. Graphs Combin. 33, 945–953 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Jensen, T.R., Toft, B.: Some Graph Coloring Problems. Wiley, New York (1995)zbMATHGoogle Scholar
  12. 12.
    Kobourov, S.G., Liotta, G., Montecchiani, F.: An annotated bibliography on 1-planarity. Comput. Sci. Rev. 25, 49–67 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Ringel, G.: Ein sechsfarbenproblem auf der Kugel. Abh. Math. Semin. Univ. Hambg. 29, 107–117 (1965)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Sun, L., Wu, J.-L.: On \((p,1)\)-total labelling of planar graphs. J. Combin. Optim. 33, 317–325 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Song, H.M., Wu, J.L., Liu, G.Z.: The equitable edge-coloring of series-parallel graphs. ICCS 2007, Part III, LNCS 4489, 457–460 (2007)Google Scholar
  16. 16.
    Wu, J.-L., Wang, P.: List-edge and list-total colorings of graphs embedded on hyperbolic surfaces. Discrete Math. 308, 6210–6215 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Zhang, X., Hou, J., Liu, G.: On total coloring of 1-planar graphs. J. Combin. Optim. 30(1), 160–173 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Zhang, X., Liu, G.: On edge colorings of 1-planar graphs without adjacent triangles. Inform. Process. Lett. 112(4), 138–142 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Zhang, X., Wu, J.-L.: On edge colorings of 1-planar graphs. Inform. Process. Lett. 111(3), 124–128 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Zhang, X., Wu, J.-L., Liu, G.: List edge and list total coloring of 1-planar graphs. Front. Math. China 7(5), 1005–1018 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Zhang, X., Yu, Y., Liu, G.: On \((p,1)\)-total labelling of 1-planar graphs. Cent. Eur. J. Math. 9(6), 1424–1434 (2011)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Japan KK, part of Springer Nature 2019

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsXidian UniversityXi’anChina
  2. 2.School of Information Science and EngineeringQufu Normal UniversityRizhaoChina

Personalised recommendations