Recursive and Cyclic Constructions for Double-Change Covering Designs

Abstract

A double-change covering design (DCCD) is a $v$-set $V$ and an ordered list $L$ 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 minimal if it has the fewest blocks possible and circular when the first and last blocks also differ by two elements. We give a recursive construction that uses 1-factorizations of complete graphs 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 minimum DCCD when $v=2k-2$. We use difference like methods to construct five infinite families of minimum circular DCCD($c\left(4k-6\right)+1,k,c^2\left(4k-6\right)+c$) when $1\le c\le 5$ for any $k\ge 3$. The recursive construction is then used to build twelve additional minimum DCCD from members of these infinite families. Finally, the difference like method is used to construct a minimum circular DCCD(61,4,366).