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The maximum genus of graphs of diameter two. (English) Zbl 0724.05021
The maximum genus $$\gamma_ M(G)$$ of a finite connected graph G is the largest genus among all closed orientable 2-manifolds in which G has a 2- cell imbedding. If $$\gamma_ M(G)$$ attains the Euler upper bound of $$(| E(G)| -| V(G)| +1)/2,$$ G is said to be upper- imbeddable. In this paper, G is assumed to be finite, connected, and of diameter two. If, in addition, G has no loops, then it is shown to be upper imbeddable. If G has loops, it is shown that $$\gamma_ M(G)$$ is at most 2 away from its Euler upper bound, if G is 2-connected; the exact value of $$\gamma_ M(G)$$ is computed, if G has connectivity 1. (Throughout the discussion, G is allowed to have multiple edges.) The main result generalizes work of V. G. Leshchenko [Properties of graphs of certain classes (Russian), Akad. Nauk Ukr. SSR, Inst. Mat., Kiev, 23-26 (1981)], who showed that every simple graph of diameter two with even Betti number is upper imbeddable.

##### MSC:
 05C10 Planar graphs; geometric and topological aspects of graph theory
##### Keywords:
maximum genus; 2-cell imbedding; upper imbeddable
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##### References:
  Behzad, M.; Chartrand, G.; Lesniak-Foster, L., Graphs and digraphs, (1979), Prindle, Weber and Schmidt Boston, MA · Zbl 0403.05027  Jungerman, M., Characterization of upper embeddable graphs, Trans. amer. math. soc., 241, 401-406, (1978) · Zbl 0379.05025  Kundu, S., Bounds on the number of disjoint spanning trees, J. combin. theory ser. B, 17, 199-203, (1974) · Zbl 0285.05113  Leshchenko, V.G., One-component imbedding of graphs of one class, (), 23-26, (Russian)  Nebeský, L., A new characterization of the maximum genus of a graph, Czechoslovak math. J., 31, 106, 604-613, (1981) · Zbl 0482.05034  Nordhaus, E.A.; Stewart, B.M.; White, A.T., On the maximum genus of a graph, J. combin. theory ser. B, 11, 258-267, (1971) · Zbl 0217.02204  Ringeisen, R.D., Survey of results on the maximum genus of a graph, J. graph theory, 3, 1-13, (1979) · Zbl 0398.05029  M. Škoviera, Decay number and the maximum genus of a graph, submitted.  Xuong, N.H., How to determine the maximum genus of a graph, J. combin. theory ser. B, 26, 217-225, (1979) · Zbl 0403.05035  Xuong, N.H., Upper embeddable graphs and related topics, J. combin. theory ser. B, 26, 226-232, (1979) · Zbl 0403.05036
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