The number of triangles in 2-factorizations

✍ Scribed by Qidi Sui; Beiliang Du

• Publisher: John Wiley and Sons
• Year: 2006
• Tongue: English
• Weight: 137 KB
• Volume: 14
• Category: Article
• ISSN: 1063-8539
• DOI: 10.1002/jcd.20088

✦ Synopsis

Abstract

Given an arbitrary 2‐factorization ${\cal F} = {F_{1},F_{2}, \cdots , F_{v - 1/2}}$ of $K_{v}$, let δ~i~ be the number of triangles contained in F~i~, and let δ = Σδ~i~. Then $\cal F$ is said to be a 2‐factorization with δ triangles. Denote by Δ(v), the set of all δ such that there exists a 2‐factorization with δ triangles. Let

where

Dejter et al. 1 proved that when v ≡ 1 or 3 (mod 6), apart from some small exceptions, and some additional 11 possible exceptions, Δ(v) = __P__Δ (v). In this paper, we consider the remaining case v ≡ 5 (mod 6). We will prove that when v ≡ 5 (mod 6), apart from some small exceptions, and some additional 9 possible exceptions, Δ (v) = P~Δ~ (v). © 2005 Wiley Periodicals, Inc. J Combin Designs 14: 277–289, 2006