Difference Methods for Double-Change Covering Designs

TL;DR

This paper introduces recursive and difference-based methods to construct minimal double-change covering designs (DCCDs), including infinite families and specific cases, advancing combinatorial design theory.

Contribution

It develops new recursive constructions and difference methods for creating minimal DCCDs, including infinite families and specific instances, expanding the known classes of these designs.

Findings

Constructed five infinite families of minimal circular DCCDs for various parameters.

Developed recursive methods to generate larger DCCDs from smaller ones.

Built twelve additional DCCDs using the proposed recursive and difference techniques.

Abstract

A \textbf{double-change covering design} (DCCD) is a $v$-set $V$ and an ordered list $\mathcal{L}$ of $b$ blocks of size $k$ where every pair from $V$ must occur in at least one block and each pair of consecutive blocks differs by exactly two elements. It is \textbf{minimal} if it has the fewest block possible and \textbf{circular} when the first and last blocks also differ by two elements. We give a recursive construction that uses 1-factorizations and expansion sets to construct a DCCD($v+\frac{v+k-2}{k-2},k,b+\frac{v}{k-2}\frac{v+k-2}{2k-4}$) from a DCCD($v,k,b$). We construct circular DCCD($2k-2,k,k-1$) and circular DCCD($2k-3,k,k-2$) from single change covering designs and determine minimal DCCD when $v=2k-2$. We use difference methods to construct five infinite families of minimal circular DCCD($c(4k-6)+1,k,c^2(4k-6)+c$) when $c\leq 5$ for any $k\geq 3$. The recursive construction is then used to build twelve additional minimal DCCD from members of these infinite families. Finally the difference method is used to construct a minimal circular DCCD(61,4,366).