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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

The genera of edge amalgamations of complete bigraphs


Author: Seth R. Alpert
Journal: Trans. Amer. Math. Soc. 193 (1974), 239-247
MSC: Primary 05C10
DOI: https://doi.org/10.1090/S0002-9947-1974-0357188-X
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.


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DOI: https://doi.org/10.1090/S0002-9947-1974-0357188-X
Keywords: Genus, Euler formula, bigraph, orientable 2-manifold
Article copyright: © Copyright 1974 American Mathematical Society