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 $\{(d_{i,1},d_{i,2},\dots ,d_{i,m})\mid i=1,2,\dots ,t\}$ such that $d_{i,1}+d_{i,2}+\cdots +d_{i,m}=0$ for $1\le i\le t$ and $\{|d_{i,j}|\mid 1\le i\le t,1\le j\le m\}=\{d,d+1,\dots ,d+mt-1\}$. 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 $m\equiv 0,2(\mathrm{mod}4)$. In the case that $m\equiv 1,3(\mathrm{mod}4)$, 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 $⟨d,d+1,\dots ,d+mt-1{⟩}_n$ for all $n\ge 2(d+mt)-1$ with $d$ and $t$ satisfying certain conditions.