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Transactions of the American Mathematical Society

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The genera of edge amalgamations of complete bigraphs

Author: Seth R. Alpert
Journal: Trans. Amer. Math. Soc. 193 (1974), 239-247
MSC: Primary 05C10
MathSciNet review: 0357188
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Abstract: If G and H are graphs, then $ G \vee H$ is defined to be a graph obtained by identifying some edge of G with some edge of H. It is shown that for all m, n, p, and q the genus $ g({K_{m,n}} \vee {K_{p,q}})$ is either $ g({K_{m,n}}) + g({K_{p,q}})$ or else $ g({K_{m,n}}) + g({K_{p,q}}) - 1$. The latter value is attained if and only if both $ {K_{m,n}}$ and $ {K_{p,q}}$ are critical in the sense that the deletion of any edge results in a graph whose genus is one less than the genus of the original graph.

References [Enhancements On Off] (What's this?)

  • [1] S. Alpert, The genera of amalgamations of graphs, Trans. Amer. Math. Soc. 178 (1973), 1-40. MR 0371698 (51:7915)
  • [2] J. Battle, F. Harary, Y. Kodama and J. W. T. Youngs, Additivity of the genus of a graph, Bull. Amer. Math. Soc. 68 (1962), 565-568. MR 27 #5247. MR 0155313 (27:5247)
  • [3] G. Ringel, Das Geschlecht des vollständigen paaren Graphen, Abh. Math. Sem. Univ. Hamburg 28 (1965), 139-150. MR 32 #6439. MR 0189012 (32:6439)
  • [4] S. Schanuel, personal communication.

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Keywords: Genus, Euler formula, bigraph, orientable 2-manifold
Article copyright: © Copyright 1974 American Mathematical Society

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