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Transactions of the American Mathematical Society

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

 

 

Neighborhood fixed pendant vertices


Authors: S. E. Anacker and G. N. Robertson
Journal: Trans. Amer. Math. Soc. 266 (1981), 115-128
MSC: Primary 05C60; Secondary 05C25
DOI: https://doi.org/10.1090/S0002-9947-1981-0613788-4
MathSciNet review: 613788
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Abstract: If $ x$ is pendant in $ G$, then $ {x^ \ast }$ denotes the unique vertex of $ G$ adjacent to $ x$. Such an $ x$ is said to be neighborhood-fixed whenever $ {x^ \ast }$ is fixed by $ A(G - x)$. It is shown that if $ G$ is not a tree and has a pendant vertex, but no *-fixed pendant vertex, then there is a subgraph $ {G^\char93 }$ of $ G$ such that for some $ y \in V({G^\char93 })$, $ O(A{({G^\char93 })_y}) \geqslant t!$ where $ t$ is the maximum number of edges in a tree rooted in $ {G^\char93 }$.


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DOI: https://doi.org/10.1090/S0002-9947-1981-0613788-4
Keywords: Reducible partitions, neighborhood fixed pendant vertices
Article copyright: © Copyright 1981 American Mathematical Society