The genus of nearly complete graphs-case 6

Abstract

The genus of a complete graph equals the least integer greater than or equal to (E-3V+6)/6, whereE andV are the numbers of edges and vertices of the graph. This paper extends the class of graphs known to have this property, concentrating on graphs whose number of vertices is congruent to 6 modulo 12.

This is a preview of subscription content, access via your institution.

References

  1. [1]

    Alpert, S. R. andGross, J. L.,Components of branched coverings of current graphs J. Combinatorial Theory (to appear).

  2. [2]

    Duke, R. andHaggard, G.,The genus of subgraphs of K 8, Israel J. Math.11, (1972), 452–455.

    Google Scholar 

  3. [3]

    Gross, J. L. andAlpert, S. R.,Branched coverings of graph imbeddings, Bull. Amer. Math. Soc.79, (1973), 942–945.

    Google Scholar 

  4. [4]

    Gross, J. L. andAlpert, S. R.,The topological theory of current graphs, J. Combinatorial Theory (B)17, (1974), 218–233.

    Google Scholar 

  5. [5]

    Ringel, G.,Bestimmung der Maximalzahl der Nachbargebiete auf nichtorientierbaren Flächen, Math. Ann.127, (1954), 181–214.

    Google Scholar 

  6. [6]

    Ringel, G. andYoungs, J. W. T.,Solution of the Heawood map-coloring problem-case 11, J. Combinatorial Theory7, (1969), 71–93.

    Google Scholar 

  7. [7]

    Youngs, J. W. T.,The Heawood map-coloring conjecture, inGraph Theory and Theoretical Physics, (F. Harary, ed.) Academic Press, London and New York, 1967, 313–354.

    Google Scholar 

  8. [8]

    Youngs, J. W. T.,Solution of the Heawood map-coloring problem-cases 3, 5, 6, and 9, J. Combinatorial Theory8, (1970), 175–219.

    Google Scholar 

Download references

Author information

Affiliations

Authors

Additional information

This research was initiated with partial support from NSF grant GJ-28403 at Columbia University and continued while the author was an IBM Postdoctoral Fellow in the Mathematical Sciences Department at the Thomas J. Watson Research Center, Yorktown Heights, N.Y. The author is also an Alfred P. Sloan Fellow.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Gross, J.L. The genus of nearly complete graphs-case 6. Aeq. Math. 13, 243–249 (1975). https://doi.org/10.1007/BF01836527

Download citation

AMS (1970) subject classifications

  • Primary 05C10, 55A15
  • Secondary 05C25, 55A10