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Size in maximal triangle-free graphs and minimal graphs of diameter 2

✍ Scribed by Curtiss Barefoot; Karen Casey; David Fisher; Kathryn Fraughnaugh; Frank Harary

Publisher: Elsevier Science
Year: 1995
Tongue: English
Weight: 290 KB
Volume: 138
Category: Article
ISSN: 0012-365X
DOI: 10.1016/0012-365x(94)00190-t

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✦ Synopsis

A triangle-free graph is maximal if the addition of any edge creates a triangle. For n ~> 5, we show there is an n-node m-edge maximal triangle-free graph if and only if it is complete bipartite or 2n-5<<.m<<.L(n-1)2/4J+l. A diameter 2 graph is minimal if the deletion of any edge increases the diameter. We show that a triangle-free graph is maximal if and only if it is minimal of diameter 2.

For n> n o where n o is a vastly huge number, Ftiredi showed that an n-node nonbipartite minimal diameter 2 graph has at most [_(n-1)2/4J+ 1 edges. We demonstrate that n o ~> 6 by producing a 6-node nonbipartite minimal diameter 2 graph with 8 edges.

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