Strong difference families over arbitrary graphs

✍ Scribed by Marco Buratti; Lucia Gionfriddo

• Publisher: John Wiley and Sons
• Year: 2008
• Tongue: English
• Weight: 200 KB
• Volume: 16
• Category: Article
• ISSN: 1063-8539
• DOI: 10.1002/jcd.20201

✦ Synopsis

Abstract

The concept of a strong difference family formally introduced in Buratti [J Combin Designs 7 (1999), 406–425] with the aim of getting group divisible designs with an automorphism group acting regularly on the points, is here extended for getting, more generally, sharply‐vertex‐transitive Γ‐decompositions of a complete multipartite graph for several kinds of graphs Γ. We show, for instance, that if Γ has e edges, then it is often possible to get a sharply‐vertex‐transitive Γ‐decomposition of K~m × e~ for any integer m whose prime factors are not smaller than the chromatic number of Γ. This is proved to be true whenever Γ admits an α‐labeling and, also, when Γ is an odd cycle or the Petersen graph or the prism T~5~ or the wheel W~6~. We also show that sometimes strong difference families lead to regular Γ‐decompositions of a complete graph. We construct, for instance, a regular cube‐decomposition of K~16m~ for any integer m whose prime factors are all congruent to 1 modulo 6. © 2008 Wiley Periodicals, Inc. J Combin Designs 16: 443–461, 2008