Abstract
Let G be a 2-edge-connected simple graph with girth g, independence number α(G), and if one of the following two conditions holds
-
(1)
α(G) ≤ 2;
-
(2)
α(G) ≥ 3, and for any three nonadjacent vertices v i (i = 1,2,3), it has
$$\sum\limits_{i=1}^{3}d_{G}(v_i) \geq \nu - 3g +7$$,
then G is upper embeddable and the lower bound v − 3g + 7 is best possible. Similarly the result for 3-edge-connected simple graph with girth g and independence number α(G) is also obtained.
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References
Bondy, J. A., Murty, U. S. R.: Graph theory with applications, Macmillan, London (1976)
Nordhaus, E. A., Stewart, B. M., White, A.T.: On the maximum genus of a graph. J. Combin. Theory 11, 258–267 (1971)
Kunda, S.: Bounds on number of disjoint spanning trees. J. Combin. Theory 17B, 199–203 (1974)
Payan, C., and Xoung, N. H.: Upper embeddability and conncetivity of graph. Discrete Math., J 27, 71–80 (1979)
Nebesky, L.: Every connected, locally connected graph is upper embeddable. J. Graph Theory 3, 197–199 (1981)
Nebesky, L.: On quasiconnected graph and their upper embeddable. Czech. Math. J., 35, 110 (1985)
Skobiera, M., Nedela, R.: The maximum genus of a graph and doubly Eulerian trail. Bulletin. U.M.I., 4B, 541–545 (1990)
Jaeger, F., Payan, C., Xuong, N. H.: A class of upper embeddable graphs. J. Graph Theory 3, 387–391 (1979)
Skoviera, M.: On the maximum of graphs of diameter two. Discrete Math 87, 175–180 (1991)
Nedela, R., Skoviera, M.: Topics in combinatorics and graph theory, R. Bodendiek and R. Henn (Eds), Phusicaverlay, Heideberg, 519–525 (1990)
Nebesky, L.: N 2-connected graph and their upper embeddability, Czech. Math. J 41, 116 (1991)
Jungerman, M.: A Characterization of upper embeddable graphs, Trans. Amer. Math. Soc., 241, 401–406 (1978)
Huang, Y. Q., Liu, Y. P.: The degree-sum of nonadjcent vertices and upper embeddability of graphs. Chin Ann of Math. 19A(5), 651–656 (1998) (in Chinese)
Chen, Y.C., Liu, Y.P.: The degree-sum of nonadacent vertices, girth and upembeddability, Graphs and Combinatorics 23, 521–527 (2007)
Xuong, N. H.: How to determine the maximum genus of a graph. J. Combinatorial Theory 26B, 217–225 (1979)
Liu, Y. P.: Embeddability in graphs. Klumer Science, London (1995)
Nebesky L.: A new characterization of the maximum genus of a graph, Czech. Math.J., 31(106), 604–613 (1981)
Huang, Y. Q., Liu, Y. P.: Maximum genus and chromatic number of graphs. Discrete Math. 271, 117–127 (2003)
Chen, Y. C., Liu, Y. P.: Maximum genus, girth and maximum nonadjacent edge set. Ars Combinatoria 79, 145–159 (2006)
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Huang Yuanqiu: Partially supported by National Science Foundation of China (No. 10771062) and Program for New Century Excellent Talents in University (No. NCET-07-0276).
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Zhangdong, O., Jing, W. & Yuanqiu, H. Upper Embeddability, Girth and the Degree-Sum of Nonadjacent Vertices. Graphs and Combinatorics 25, 253–264 (2009). https://doi.org/10.1007/s00373-008-0837-1
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DOI: https://doi.org/10.1007/s00373-008-0837-1