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Graphs with Maximal Even Girth

Published online by Cambridge University Press:  20 November 2018

A. Gewirtz
A. Gewirtz*
Affiliation:
Brooklyn College, Brooklyn, New York
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In this paper we examine the class G of simple undirected, connected graphs of diameter d > 1, girth 2d, and for any g ɛ G, if a pair of nodes are at distance d from each other, then that pair of nodes is connected by t distinct paths of length d, t > 1. (The girth of g is the length of the smallest circuit in g.)

We establish, in § 2, that for all g ɛ G, g is regular.

We establish necessary conditions for the existence of elements of G. If g ɛ G, we adopt the notation g = g(d, t, v, n), where v is the valence of g and n is the number of nodes. It is of course possible for g, h ɛ G, g ≠ h, and for given d, t, v, n to have both g(d, t, v, n) and h(d, t, v, n).

In particular, we show that if d = 2, t ≠ 2, 4 or 6, then there is at most a finite number of graphs with a particular given t value.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 21 , 1969 , pp. 915 - 934
Copyright
Copyright © Canadian Mathematical Society 1969

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