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A method to construct 1‐rotational factorizations of complete graphs and solutions to the oberwolfach problem
Author(s) -
McGinnis Daniel,
Poimenidou Eirini
Publication year - 2019
Publication title -
journal of combinatorial designs
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.618
H-Index - 34
eISSN - 1520-6610
pISSN - 1063-8539
DOI - 10.1002/jcd.21643
Subject(s) - mathematics , conjugacy class , factorization , combinatorics , finite group , pairwise comparison , pure mathematics , discrete mathematics , group (periodic table) , algorithm , chemistry , statistics , organic chemistry
The concept of a 1‐rotational factorization of a complete graph under a finite group G was studied in detail by Buratti and Rinaldi. They found that if G admits a 1‐rotational 2‐factorization, then the involutions of G are pairwise conjugate. We extend their result by showing that if a finite group G admits a 1‐rotational k ‐factorization with k = 2 n m even and m odd, then G has at most m ( 2 n − 1 ) conjugacy classes containing involutions. Also, we show that if G has exactly m ( 2 n − 1 ) conjugacy classes containing involutions, then the product of a central involution with an involution in one conjugacy class yields an involution in a different conjugacy class. We then demonstrate a method of constructing a 1‐rotational 2 n ‐factorization under G × Z n given a 1‐rotational 2‐factorization under a finite group G . This construction, given a 1‐rotational solution to the Oberwolfach problem O P ( a ∞ , a 1 , a 2… , a n ) , allows us to find a solution to O P ( 2 a ∞ − 1 , 2a 1 , 2a 2… , 2a n ) when the a i ’s are even ( i ≠ ∞ ), and O P ( p ( a ∞ − 1 ) + 1 , pa 1 , pa 2… , pa n ) when p is an odd prime, with no restrictions on the a i ’s.