Langford-type Difference Sets for Cycle Systems

Abstract

A Langford-type \( m \)-tuple difference set of order \( t \) and defect \( d \) is a set of \( t \) \( m \)-tuples \( \{(d_{i,1}, d_{i,2}, \ldots, d_{i,m}) \mid i = 1, 2, \ldots, t\} \) such that \( d_{i,1} + d_{i,2} + \cdots + d_{i,m} = 0 \) for \( 1 \leq i \leq t \) and \( \{|d_{i,j}| \mid 1 \leq i \leq t, 1 \leq j \leq m\} = \{d, d+1, \ldots, d+mt-1\} \). In this paper, we give necessary and sufficient conditions on \( t \) and \( d \) for the existence of a Langford-type \( m \)-tuple difference set of order \( t \) and defect \( d \) when \( m \equiv 0, 2 \pmod{4} \). In the case that \( m \equiv 1, 3 \pmod{4} \), we provide sufficient conditions for the existence of a Langford-type \( m \)-tuple difference set of order \( t \) and defect \( d \) when \( d \) is at most about \( t/2 \). Using these results, we obtain cyclic \( m \)-cycle systems of the circulant graph \( \langle{d, d+1, \ldots, d+mt-1}\rangle_n \) for all \( n \geq 2(d+mt)-1 \) with \( d \) and \( t \) satisfying certain conditions.