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Longest Cycles in 2-Connected Graphs with Prescribed Maximum Degree

Published online by Cambridge University Press:  20 November 2018

J. A. Bondy  and
R. C. Entringer
J. A. Bondy
Affiliation:
University of Waterloo, Waterloo, Ontario
R. C. Entringer
Affiliation:
University of New Mexico, Albuquerque, New Mexico
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The relationship between the lengths of cycles in a graph and the degrees of its vertices was first studied in a general context by G. A. Dirac. In [5], he proved that every 2-connected simple graph on n vertices with minimum degree d contains a cycle of length at least min{2d, n};. Dirac's theorem was subsequently strengthened in various directions in [7], [6], [13], [12], [2], [1], [11], [8], [14], [15] and [16].

Our aim here is to investigate another aspect of this relationship, namely how the lengths of the cycles in a 2-connected graph depend on the maximum degree. Let us denote by ƒ(n, d) the largest integer k such that every 2-connected simple graph on n vertices with maximum degree d contains a cycle of length at least k. We prove in Section 2 that, for d ≧ 3 and n ≧ d + 2,

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 32 , Issue 6 , 01 December 1980 , pp. 1325 - 1332
Copyright
Copyright © Canadian Mathematical Society 1980

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