Let a random graph G be constructed by adding random edges one by one, starting with n isolated vertices. We show that with probability going to one as n goes to infinity, when G first has minimum degree two, it has at least (log n)('-')" distinct hamilton cycles for any fixed E > 0.
The number of cycles in a hamilton graph
β Scribed by Yongbing Shi
- Publisher
- Elsevier Science
- Year
- 1994
- Tongue
- English
- Weight
- 523 KB
- Volume
- 133
- Category
- Article
- ISSN
- 0012-365X
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We consider the standard random geometric graph process in which n vertices are placed at random on the unit square and edges are sequentially added in increasing order of edge-length. For fixed k β₯ 1, we prove that the first edge in the process that creates a k-connected graph coincides a.a.s. with
## UNIVERSIW OF WATERLOO ' The research reported here has been sponsored by the Canadian Commonwealth Association.
## Abstract Let __G__ be a graph on __p__ vertices with __q__ edges and let __r__β=β__q__βββ__p__β=β1. We show that __G__ has at most ${15\over 16} 2^{r}$ cycles. We also show that if __G__ is planar, then __G__ has at most 2^__r__βββ1^β=β__o__(2^__r__βββ1^) cycles. The planar result is best possib