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Finding \(\Delta (\Sigma )\) for a Surface \(\Sigma \) of Characteristic \(-6\) and \(-7\)

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Abstract

For each surface \(\Sigma \), we define \(\Delta (\Sigma ) =\) max\(\{\Delta (G)|\,G\) is a class two graph with maximum degree \(\Delta (G)\) that can be embedded in \(\Sigma \}\). Hence Vizing’s Planar Graph Conjecture can be restated as \(\Delta (\Sigma )=5\) if \(\Sigma \) is a sphere. For a surface \(\Sigma \), if \(\chi (\Sigma ) \in \{-5, -4, \dots , 0\}\), \(\Delta (\Sigma )\) is already known. In this paper, we show that \(\Delta (\Sigma )=10\) if \(\Sigma \) is a surface of characteristic \(\chi (\Sigma ) \in \{-6, -7\}\).

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References

  1. Beineke, L.W., Fiorini, S.: On small graphs critical with respect to edge-colourings. Discrete Math. 16, 109–121 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bokal, D., Brinkmann, G., Grünewald, S.: Chromatic-index-critical graphs of orders 13 and 14. Discrete Math. 300, 16–29 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  3. Brinkmann, G., Steffen, E.: Chromatic-index-critical graphs of orders 11 and 12. Eur. J. Combin. 19, 889–900 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  4. Chetwynd, A.G., Yap, H.P.: Chromatic index critical graphs of order 9. Discrete Math. 47, 23–33 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  5. Gross, J.L., Tucker, T.W.: Topological Graph Theory. Dover, New York (2012)

    MATH  Google Scholar 

  6. Hind, H., Zhao, Y.: Edge colorings of graphs embeddable in a surface of low genus. Discrete Math. 190, 107–114 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  7. Luo, R., Miao, L.Y., Zhao, Y.: The size of edge chromatic critical graphs with maximum degree 6. J. Graph Theory 60, 149–171 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  8. Luo, R., Zhao, Y.: Finding the exact bound of the maximum degree of class two graphs embeddable in a surface of characteristic \(\epsilon \in \{-1,-2,-3\}\). J. Combin. Theory Ser. B 98, 707–720 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  9. Luo, R., Zhao, Y.: Finding \(\Delta (\Sigma )\) for a surface \(\Sigma \) of characteristic \(\chi (\Sigma ) =-5\). J. Graph Theory 68, 148–168 (2011)

    Article  MathSciNet  Google Scholar 

  10. Luo, R., Miao, Z.K., Zhao, Y.: Finding \(\Delta (\Sigma )\) for a surface \(\Sigma \) of characteristic \(\chi (\Sigma ) =-4\). J. Graph Theory (2011). doi:10.1002/jgt.21997

    MathSciNet  Google Scholar 

  11. Mel’nikov, L.S., The chromatic class and location of a graph on a closed surface, Mat. Zametki 7 (1970) 671–681 [Math. Notes 7, 405–411 (1970)]

  12. Sanders, D., Zhao, Y.: Planar graphs of maximum degree seven are class I. J. Combin. Theory Ser. B 83, 201–212 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  13. Sanders, D., Zhao, Y.: Coloring edges of graphs embedded in a surface of characteristic zero. J. Combin. Theory Ser. B 87, 254–263 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  14. Steibitz, M., Scheide, D., Toft, B., Favrholdt, L.: Graph Edge Coloring. Wiley, New York (2012)

    MATH  Google Scholar 

  15. Vizing, V.G.: Critical graphs with a given chromatic class (Russian). Diskret. Analiz. 5, 9–17 (1965)

    MathSciNet  Google Scholar 

  16. Vizing, V.G., Some unsolved problems in graph theory, Uspekhi Mat. Nauk 23 (1968) 117–134 [Russian Math. Surveys 23, 125–142 (1968)]

  17. Yan, Z., Zhao, Y.: Edge colorings of embedded graphs. Graphs Combin. 16, 245–256 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  18. Zhang, L.: Every planar graph with maximum degree 7 is of class 1. Graphs Combin. 16, 467–495 (2000)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

R. Luo was supported by NSFC under Grant Numbers 11401003, and Z. Miao was supported by NSFC under Grant Numbers 11171288 and 11571149.

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Authors and Affiliations

  1. Department of Mathematics, West Virginia University, Morgantown, WV, 26506-6310, USA

    Katie Horacek & Rong Luo

  2. School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou, 221116, Jiangsu, China

    Zhengke Miao

  3. Department of Mathematics, University of Central Florida, Orlando, FL, 32816-1364, USA

    Yue Zhao

Authors
  1. Katie Horacek
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  2. Rong Luo
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  3. Zhengke Miao
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  4. Yue Zhao
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Correspondence to Zhengke Miao.

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Horacek, K., Luo, R., Miao, Z. et al. Finding \(\Delta (\Sigma )\) for a Surface \(\Sigma \) of Characteristic \(-6\) and \(-7\) . Graphs and Combinatorics 33, 929–944 (2017). https://doi.org/10.1007/s00373-017-1780-9

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  • DOI: https://doi.org/10.1007/s00373-017-1780-9

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