Minimum Degree and Even Cycle Lengths
Document Type
Article
Publication Date
2016
Publication Title
Bulletin of the Institute of Combinatorics and its Applications
Abstract
A classic result of Dirac states that if G is a 2-connected graph of order n with minimum degree δ ≥ 3, then G contains a cycle of length at least min{n, 2δ}. In this paper, we consider the problem of determining the number of different odd or even cycle lengths that must exist under the minimum degree condition. We conjecture that there are δ − 1 even cycles of different lengths, and when G is nonbipartite, that there are δ− 1 odd cycles of different lengths. We prove this conjecture when δ = 3. Related results concerning the number of different even cycle lengths supporting the conjecture are also included. In particular, we show that there are always at least (δ − 1)/2 even cycles of different lengths.
Recommended Citation
Faudree, Ralph J., Ronald Gould, Michael S. Jacobson, Colton Magnant.
2016.
"Minimum Degree and Even Cycle Lengths."
Bulletin of the Institute of Combinatorics and its Applications, 77: 59-70.
source: https://pdfs.semanticscholar.org/855d/f9f89c2d24e2f87361fc8b6a08d87c40c437.pdf
https://digitalcommons.georgiasouthern.edu/math-sci-facpubs/587
