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On 2-cell embeddings of graphs with minimum numbers of regions. (English) Zbl 0586.05015
Let $$\gamma_ M(G)$$ denote the maximum genus of a graph G and let $$\beta$$ (G) denote its Betti number. A graph G is upper embeddable provided that $$\gamma_ M(G)=\lfloor \beta (G)/2\rfloor$$. Given G and a spanning subgraph J, the author derives necessary and sufficient conditions for when there exists an upper embeddable H with $$J\subset H\subset G$$ (supposing that J is connected). He also derives necessary and sufficient conditions for when every such H is upper embeddable (supposing that G is connected).
Reviewer: D.S.Archdeacon

##### MSC:
 05C10 Planar graphs; geometric and topological aspects of graph theory
##### Keywords:
upper embeddable graph; maximum genus
Full Text:
##### References:
 [1] M. Behzad G. Chartrand, and L. Lesniak-Foster: Graphs & Digraphs. Prindle, Weber & Schmidt, Boston 1979. · Zbl 0403.05027 [2] F. Harary: Graph Theory. Addison-Wesley, Reading (Mass.) 1969. · Zbl 0196.27202 [3] N. P. Homenko N. A. Ostroverkhy, and V. A. Kusmenko: The maximum genus of a graph. (in Ukrainian, English summary), $$\varphi$$-peretvorennya grafiv (N. P. Homenko IM AN URSR, Kiev 1973, pp. 180-210. [4] M. Jungerman: A characterization of upper embeddable graphs. Trans. Amer. Math. Soc. 241 (1978), 401-406. · Zbl 0379.05025 [5] L. Nebeský: A new characterization of the maximum genus of a graph. Czechoslovak Math. J. 31 (106) (1981), 604-613. · Zbl 0482.05034 [6] L. Nebeský: A note on upper embeddable graphs. Czechoslovak Math. J. 33 (108) (1983), 37-40. [7] L. Nebeský: On a diffusion of a set of vertices in a connected graph. Graphs and Other Combinatorial Topics (Proc. Third Czechoslovak Symp. Graph Theory held in Prague, 1982) (M. Fiedler, Teubner-Texte zur Mathematik, Band 59, Teubner, Leipzig 1983, pp. 200-203. [8] E. A. Nordhaus R. D. Ringeisen B. M. Stewart, and A. T. White: A Kuratowski-type theorem for the maximum genus of a graph. J. Combinatorial Theory 12 B (1972), 260-267. · Zbl 0217.02301 [9] G. Ringel: The combinatorial map color theorem. J. Graph Theory 1 (1977), 141 - 155. · Zbl 0386.05030 [10] N. H. Xuong: How to determine the maximum genus of a graph. J. Combinatorial Theory 26 B (1979), 217-225. · Zbl 0403.05035
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