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Edge-Maximal Graphs on Surfaces

Published online by Cambridge University Press:  20 November 2018

Colin McDiarmid  and
David R. Wood
Colin McDiarmid
Affiliation:
Department of Statistics, University of Oxford, United Kingdom email: cmcd@stats.ox.ac.uk
David R. Wood
Affiliation:
Department of Statistics, University of Oxford, United Kingdom email: cmcd@stats.ox.ac.uk
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Abstract

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We prove that for every surface $\Sigma $ of Euler genus $g$, every edge-maximal embedding of a graph in $\Sigma $ is at most $O(g)$ edges short of a triangulation of $\Sigma $. This provides the first answer to an open problem of Kainen (1974).

Keywords

05C10graphsurfaceembedding
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 70 , Issue 4 , 01 August 2018 , pp. 925 - 942
Copyright
Copyright © Canadian Mathematical Society 2018

References

[1] Archdeacon, D., The nonorientable genus is additive. J. Graph Theory 10(1986), 363–383. http://dx.doi.org/10.1002/jgt.3190100313 Google Scholar
[2] Harary, F., Kainen, P. C., Schwenk, A. J., and White, A. T., A maximal toroidal graph which is not a triangulation. Math. Scand. 33(1973), 108–112. http://dx.doi.Org/10.7146/math.scand.a-11476 Google Scholar
[3] Joret, G. and Wood, D. R., Irreducible triangulations are small. J. Combin. Theory Ser. B 100(2010), no. 5, 446–455. http://dx.doi.Org/10.1016/j.jctb.2010.01.004 Google Scholar
[4] Kainen, P. C., Some recent results in topological graph theory. In: Graphs and combinatorics (Proc. Capital Conf., George Washington Univ., Washington D.C., 1973), Lecture Notes in Math., 406, Springer, Berlin, 1974, pp. 76–108.Google Scholar
[5] Kostochka, A. V., Lower bound of the Hadwiger number of graphs by their average degree. Combinatorica 4(1984), no. 4, 307–316. http://dx.doi.org/10.1007/BF02579141 Google Scholar
[6] McDiarmid, C. and Przykucki, M., On the purity of minor-closed classes of graphs. 2016. arxiv:1 608.08623Google Scholar
[7] Miller, G. L., An additivity theorem for the genus of a graph. J. Combin. Theory Ser. B 43(1987), no. 1, 25–47. http://dx.doi.org/10.1016/0095-8956(87)90028-1 Google Scholar
[8] Mohar, B. and Thomassen, C., Graphs on surfaces. Johns Hopkins Studies in the Mathematical Sciences, Johns Hopkins University Press, Baltimore, MD, 2001.Google Scholar
[9] Nakamoto, A. and Ota, K., Note on irreducible triangulations of surfaces. J. Graph Theory 20(1995), 227–233. http://dx.doi.org/10.1002/jgt.3190200211 Google Scholar
[10] Richter, R. B., On the Euler genus of a 2-connected graph. J. Combin. Theory Ser. B 43(1987), 60–69. http://dx.doi.org/10.1016/0095-8956(87)90030-X Google Scholar
[11] Ringel, G., Das Geschlecht des vollstaändigen paaren Graphen. Abh. Math. Sem. Univ. Hamburg 28(1965, 139–150. http://dx.doi.org/10.1007/BF02993245 Google Scholar
[12] Ringel, G., Der vollstaändige paare Graph auf nichtorientierbaren Flächen. J. Reine Angew. Math. 220(1965), 88–93. http://dx.doi.Org/10.1515/crll.1965.220.88 Google Scholar
[13] Thomason, A., An extremal function for contractions of graphs. Math. Proc. Cambridge Philos. Soc. 95(1984), 261–265. http://dx.doi.org/10.1017/S0305004100061521 Google Scholar
[14] Thomason, A., The extremal function for complete minors. J. Combin. Theory Ser. B 81(2001), 318–338. http://dx.doi.Org/10.1006/jctb.2000.2013 Google Scholar
[15] Thomassen, C., Embeddings of graphs with no short noncontractible cycles. J. Combin. Theory Ser. B 48(1990), 155–177. http://dx.doi.org/10.1016/0095-8956(90)90115-C Google Scholar
[16] Wagner, K., Über eine Eigenschaft der ebene Komplexe. Math. Ann. 114(1937), 570–590. http://dx.doi.org/10.1007/BF01594196 Google Scholar