Premium
Solution to the outstanding case of the spouse‐loving variant of the Oberwolfach problem with uniform cycle length
Author(s) -
Shanmuga Vadivu Andiyappan,
Panneerselvam Lakshmanan,
Muthusamy Appu
Publication year - 2021
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.21759
Subject(s) - mathematics , conjecture , combinatorics , integer (computer science) , spouse , order (exchange) , factorization , graph , discrete mathematics , algorithm , law , computer science , finance , economics , programming language , political science
Abstract Let K n + I denote the complete graph of even order with a 1‐factor duplicated. The spouse‐loving variant of the Oberwolfach Problem, denoted O P + ( m 1 , m 2 , … , m t ) , asks for the existence of a 2‐factorization of K n + I in which each 2‐factor consists of cycles of length m i , for all i , 1 ≤ i ≤ t , such that n = m 1 + m 2 + ⋯ + m t . If m 1 = m 2 = ⋯ = m t = m , then the problem is denoted by O P + ( n ; m ) . In this paper, we construct a solution to O P + ( 4 m ; m ) when m ≥ 5 is an odd integer. This completes the proof of the conjecture posed by Bolohan et al. In addition, we find a solution to O P + ( 3 , m ) when m ≥ 5 is an odd integer.