Abstract
Let G be a connected graph of order n and girth g. If d G (u) + d G (v) ≥ n − 2g + 5 for any two non-adjacent vertices u and v, then G is up-embeddable. Further more, the lower bound is best possible. Similarly the result of k-edge connected simple graph with girth g is also obtained, k = 2,3.
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References
Bondy, J.A., Murty, U.S.R.: Graph Theory with Applications, Macmillan, London 1979
Huang, Y.Q., Liu, Y.P.: An improvement of a theorem on the maximum genus for graphs, Math. Appl. 11, 109–112 (1998)
Huang, Y.Q., Liu. Y.P.: The degree-sum of nonadjacent vertices and up-embeddability of graphs, Chinese J. Contemp. Math. 19, 651–656 (1998) (in Chinese)
Jaeger, F., Payan, C., Xuong, N.H.: A class of upper embeddable graphs, J. Graph Theory 3, 387–391 (1979)
Jungerman, M.: A characterization of upper embeddable graphs, Trans. Am. Math. Soc. 241, 401–406 (1987)
Khomendo, N.P., Glukhov, A.D.: On upper embeddable graphs. Graph Theory, (N.P. Khomenko, ed) Izd. Akad. Nauk. Ukrain. SSR. Kiev., pp 85–87 (1977) (in Russian)
Kundu, S.: Bounds on number of disjoint spanning trees, J. Combin. Theory B 17, 199–203 (1974)
Liu, Y.P.: The maximum orientable genus of a graph, Scientia Sinica, Special Issue on Math. II, 41–55 (1979)
Liu, Y.P.: Embeddability in Graphs, Kluwer, Dordrecht/Boston/London, 1995
Nebesky, L.: Every connected, locally connected graph is upper embeddable, J. Graph Theory 3, 197–199 (1981)
Nebesky, L.: On locally quasiconnected graph and their upper embeddability, Czech. Math. J. 35, 162–166 (1985)
Nebesky, L.: N2-connected graph and their upper embeddability, Czech. Math. J. 41, 731–735 (1991)
Nebesky, L.: A new characterization of the maximum genus of a graph, Czech. Math. J. 31, 604–613 (1981)
Nedela, R., Skoviera, M.: Topics in combinatorics and graph theory. R. Bodendiek and R. Henn (eds), Physicaverlay, Heideberg, pp 519–525 (1990)
Nordhaus, E.A., Stewart, B.M., White, A.T.: On the maximum genus of a graph, J. Combin. Theory 11, 285–267 (1971)
Nordhaus, E.A., Ringeisen, R.D., Stewart, B.M., White, A.T.: A Kuratowski type theorem for the maximum genus of a graph, J. Combin. Theory B 12, 260–267 (1972)
Skoviera, M.: On the maximum genus of graphs of diameter two, Discrete Math. 87, 175–180 (1991)
Skoviera, M., Nedela, R.: The maximum genus of a graph and doubly Eulerian trail, Bulletin, U. M. I. 4B, 541–545 (1990)
White, A.T.: Graphs, Groups and Surfaces, 2nd edn, Elsevier, Amsterdam, 1984
Xuong, N.H.: How to determine the maximum genus of a graph, J. Combin. Theory B 26, 217–225 (1979)
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Partially supported by the Postdoctoral Seience Foundation of Central South University and NNSFC under Grant No. 10751013.
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Chen, Y., Liu, Y. Up-Embeddability of a Graph by Order and Girth. Graphs and Combinatorics 23, 521–527 (2007). https://doi.org/10.1007/s00373-007-0746-8
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DOI: https://doi.org/10.1007/s00373-007-0746-8