When Hamilton circuits generate the cycle space of a random graph
Abstract
If eps > 0 and p >= n^{-1/2 + eps}, in a binomial random graph G(n,p) a.a.s. the set of cycles which can be constructed as a symmetric difference of Hamilton circuits is as large as parity by itself permits (all cycles if n is odd, all even cycles if n is even). Moreover, every p which ensures the above property a.a.s. must necessarily be such that for any constant c>0, eventually p >= (log n + 2 log log n + c)/n. So, whatever the smallest sufficient p for an a.a.s. Hamilton-generated cycle space might be, it does not coincide with the threshold for hamiltonicity of G(n,p).
Keywords
Cite
@article{arxiv.1303.0026,
title = {When Hamilton circuits generate the cycle space of a random graph},
author = {Peter C. Heinig},
journal= {arXiv preprint arXiv:1303.0026},
year = {2013}
}
Comments
8 pages, 1 figure; added proof that any p which a.a.s. guarantees a non-trivial cycle space generated by Hamilton circuits must be significantly larger than the threshold for hamiltonicity of G(n,p)