Constructions for Some Classes of Neighbor Designs

Abstract

Rees [2] introduced the concept and name of "neighbor" designs. The problem can be described as that of arranging $v$ kinds of objects on $b$ plates each containing $k$ objects in a loop such that every object on a plate has two neighbors. The requirements are that each object appears $r$ times (but not necessarily on $r$ different plates) and is a neighbor of every other object exactly $\lambda$ times. This paper constructs neighbor designs with parameters as follows: $(i) k > 2:\quad v = 2k + 1, \lambda = 1$ $(ii) k \equiv 0 (\mod 2) > 2:\quad v = 2^i k + 1, i = 1,2,\cdots, \lambda = 1$ $(iii) k \equiv 0 (\mod 4):\quad v = 2mk + 1, m = 1,2,\cdots, \lambda = 1.$