Bipartite variation of the cheesecake factory problem: [formula omitted]-factorization of [formula omitted

Abstract

For k ≥ 2 l + 1 ≥ 3 , let H ( k , 2 l + 1 ) be a bipartite graph with bipartition X = { x 1 , x 2 , … , x k } , Y = { y 1 , y 2 , … , y k } and edge set { x i y i ± j ∣ 1 ≤ i ± j ≤ k , i = 1 , 2 , … , k ; j = 0 , 1 , 2 , … , l } . In 2009, Dalibor Froncek raised the Rectangular Table Negotiation Problem: In graph theoretical terms it is equivalent to finding an m H ( k , 2 l + 1 ) -factorization of K n , n , where n = m k . Also he answered the above problem for H ( k , 3 ) when k is odd and left open the remaining cases. In this paper, we show that the necessary conditions n = m k and m ≡ 0 ( mod ε ( k , l ) d ) , where ε ( k , l ) = k ( 2 l + 1 ) − l ( l + 1 ) = the number of edges in H ( k , 2 l + 1 ) and d = gcd ( k 2 , ε ( k , l ) ) for the existence of m H ( k , 2 l + 1 ) -factorization of K n , n are also sufficient when d = 1 , 2 , l , or l + 1 ≡ 0 ( mod d ) . In fact our results partially answer the Rectangular Table Negotiation Problem and also deduce the result of Froncek as a corollary.