We prove that if m is odd then a partial m-cycle system on n vertices can be embedded in an m-cycle system on at most m((m -2)n(n -1) + 2n + 1) vertices and that a partial weak Steiner m-cycle system on n vertices can be embedded in an m-cycle system on m(2n + 1) vertices.
Small embeddings for partial triple systems of odd index
β Scribed by Darryn Bryant; Geoffrey Martin
- Book ID
- 113698796
- Publisher
- Elsevier Science
- Year
- 2012
- Tongue
- English
- Weight
- 365 KB
- Volume
- 119
- Category
- Article
- ISSN
- 0097-3165
No coin nor oath required. For personal study only.
π SIMILAR VOLUMES
## Abstract A wellβknown, and unresolved, conjecture states that every partial Steiner triple system of order __u__ can be embedded in a Steiner triple system of order Ο for all Ο ββ‘ 1 or 3, (mod 6), Ο ββ₯β2uβ+β1. However, some partial Steiner triple systems of order __u__ can be embedded in Steiner t
## Abstract It has been conjectured that any partial 5βcycle system of order __u__ can be embedded in a 5βcycle system of order __v__ whenever **__v__**β₯**3****__u__/****2+1** and **__v__**β‘**1****,****5 (mod 10)**. The smallest known embeddings for any partial 5βcycle system of order __u__ is **10
Let m = 2k. We show that for some 0 β€ < 1, a partial directed m-cycle system of order n can be embedded in a directed m-cycle system of order (mn)/2 + (2m 2 + 1) (8n + 1)/4 + 4m 3 2 + 4m + 1/2. For fixed m, this is asymptotic in n to (mn)/2 and so for large n is roughly one-fourth the best known bou
## Abstract Lindner's conjecture that any partial Steiner triple system of order __u__ can be embedded in a Steiner triple system of order __v__ if $v\equiv 1,3 \; ({\rm mod}\; 6)$ and $v\geq 2u+1$ is proved. Β© 2008 Wiley Periodicals, Inc. J Combin Designs 17: 63β89, 2009