A. Burgess | Searcharxiv

arXiv2025

On circular external difference families

Circular external difference families (CEDFs) are a recently-introduced variation of external difference families (EDFs) with applications to non-malleable threshold schemes: a $(v,m,\ell,1)$-CEDF is an $m$-sequence $(A_0, \ldots, A_{m-1})$ of $\ell$-subsets of an additive group $G$ of order $v$ such that $G\setminus\{0\}$ equals the multiset of all differences $a-a'$, with $(a,a')\in A_{i+1}\times A_{i}$ for some $i \in \mathbb{Z}_m$. When $G$ is the cyclic group, we speak of a cyclic CEDF. The existence of cyclic $(v,m,\ell,1)$-CEDFs is well understood when $m$ is even, while nonexistence is known when both $m$ and $\ell$ are odd. However, the case where $m$ is odd and $\ell$ is even has only been resolved in a few special cases. In this paper, we address this gap by constructing cyclic $(v,m,\ell,1)$-CEDFs for any odd $m>1$ when $\ell=2$, and for any even $\ell \ge 2$ when $m=3$. Notably, the latter result relies on the existence of a suitable tiling of the multiplicative semigroup of $\mathbb{Z}_v\setminus\{0\}$. Moreover, noting that every $(v,3,\ell,1)$-CEDF produces a $(v,3,\ell,2)$-EDF, we completely solve the existence problem for $(3\ell^2+1,3,\ell,2)$-EDFs over an abelian group. Our approach is based on representing the blocks as arithmetic progressions and analyzing their step patterns. We present two different ways to construct cyclic $(v,m,2,1)$-CEDFs for every odd $m>1$; their step patterns show that the resulting CEDFs are inequivalent. Many additional inequivalent CEDFs are obtained by translating suitable subsets within the CEDF.

math.CO↗

arXiv2015

On the Hamilton-Waterloo Problem with odd orders

Given non-negative integers $v, m, n, α, β$, the Hamilton-Waterloo problem asks for a factorization of the complete graph $K_v$ into $α$ $C_m$-factors and $β$ $C_n$-factors. Clearly, $v$ odd, $n,m\geq 3$, $m\mid v$, $n\mid v$ and $α+β= (v-1)/2$ are necessary conditions. To date results have only been found for specific values of $m$ and $n$. In this paper we show that for any $m$ and $n$ the necessary conditions are sufficient when $v$ is a multiple of $mn$ and $v>mn$, except possibly when $β=1$ or 3, with five additional possible exceptions in $(m,n,β)$. For the case where $v=mn$ we show sufficiency when $β> (n+5)/2$ except possibly when $(m,α) = (3,2)$, $(3,4)$, with seven further possible exceptions in $(m,n,α,β)$. We also show that when $n\geq m\geq 3$ are odd integers, the lexicographic product of $C_m$ with the empty graph of order $n$ has a factorization into $α$ $C_m$-factors and $β$ $C_n$-factors for every $0\leq α\leq n$, $β= n-α$, except possibly when $α= 2,4$, $β= 1, 3$, with three additional possible exceptions in $(m,n,α)$.

math.CO↗