Conditions ensuring starter induced 1-factorizations are not isomorphic

✍ Scribed by E. Ihrig

• Publisher: John Wiley and Sons
• Year: 2003
• Tongue: English
• Weight: 166 KB
• Volume: 11
• Category: Article
• ISSN: 1063-8539
• DOI: 10.1002/jcd.10023

✦ Synopsis

Abstract

In general, it is difficult to determine whether two starter induced 1‐factorizations of K~2__n__~ are isomorphic. However, when one of the 1‐factorizations has a unique starter group to within conjugacy, we show that two starter induced 1‐factorizations on K~2__n__~ are isomorphic, if and only if, the corresponding starters are isomorphic. Two starters are isomorphic if there is a group isomorphism between the respective starter groups which takes one starter to the other. It is relatively easy to check whether starters are isomorphic in many examples. The difficulty comes in showing there is a unique starter group to within conjugacy. A number of sufficient conditions for this are given; one such condition is the assumption that the 1‐factorization be irreducible, a condition which applies to all almost perfect 1‐factorizations. Also, generalized Mullin Nemeth starters are introduced and discussed in this context. © 2003 Wiley Periodicals, Inc. J Combin Designs 11: 124–143, 2003; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/jcd.10023