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Transactions of the American Mathematical Society

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

 
 

 

The genera of amalgamations of graphs


Author: Seth R. Alpert
Journal: Trans. Amer. Math. Soc. 178 (1973), 1-39
MSC: Primary 05C10
DOI: https://doi.org/10.1090/S0002-9947-1973-0371698-X
MathSciNet review: 0371698
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Abstract: If $ p \leq m$, n then $ {K_m}{ \vee _{{K_p}}}{K_n}$ is the graph obtained by identify ing a copy of $ {K_p}$ contained in $ {K_m}$ with a copy of $ {K_p}$ contained in $ {K_n}$ . It is shown that for all integers $ p \leq m$, n the genus $ g({K_m}{ \vee _{{K_p}}}{K_n})$ of $ {K_m}{ \vee _{{K_p}}}{K_n}$ is less than or equal to $ g({K_m}) + g({K_n})$. Combining this fact with the lower bound obtained from the Euler formula, one sees that for $ 2 \leq p \leq 5,g({K_m}{ \vee _{{K_p}}}{K_n})$ is either $ g({K_m}) + g({K_n})$ or else $ g({K_m}) + g({K_n}) - 1$. Except in a few special cases, it is determined which of these values is actually attained.


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DOI: https://doi.org/10.1090/S0002-9947-1973-0371698-X
Keywords: Orientable 2-manifold, genus, Euler formula, Complete Graph Theorem
Article copyright: © Copyright 1973 American Mathematical Society

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