Partial triple systems and edge colourings

✍ Scribed by Alan Hartman

• Book ID: 103056658
• Publisher: Elsevier Science
• Year: 1986
• Tongue: English
• Weight: 715 KB
• Volume: 62
• Category: Article
• ISSN: 0012-365X
• DOI: 10.1016/0012-365x(86)90117-2

✦ Synopsis

A partial triple system of order v, PT(v), is a pair (V, B) where V is a v-set, and B is a collection of 3-subsets of V (called triples) such that each 2-subset of V is contained in at most one triple. A maximum partial triple system of order v, MPT(v), is a PT(v), (V, B), such that for any other PT(v), (V, C), we have [C I ~< IBI. Several authors have considered the problem of embedding PT(v) and MPT(v) in systems of higher order. We complete the proof, begun by Mendelsohn and Rosa [6], that an MPT(u) can be embedded in an MPT(v) where v is the smallest value in each congruence class rood 6 with v ~> 2u. We also consider a general problem concerning transversals of minimum edge-colourings of the complete graph.

Lemma 1. Let u > 6, and let (all, A) be an MPT(u) embedded in an MPT(v). Then v >>-2u.