Constructing R-sequencings and terraces for groups of even order

The problem of finding R-sequencings for abelian groups of even orders has been reduced to that of finding R*-sequencings for abelian groups of odd orders except in the case when the Sylow 2-subgroup is a non-cyclic non-elementary-abelian group of order 8. We partially address this exception, including all instances when the group has order 8t for t congruent to 1, 2, 3 or 4 (mod7). As much is known about which odd-order abelian groups are R*-sequenceable, we have constructions of R-sequencings for many new families of abelian groups. The construction is generalisable in several directions, leading to a wide array of new R-sequenceable and terraceable non-abelian groups of even order

R-sequenceability and R*-sequenceability of abelian 2-groups

A group of order n is said to be R-sequenceable if the nonidentify elements of the group can be listed in a sequence a1,a2,...,an-1 such that the quotients a-11a2,a-12a3,...,a-1n-2an-1,a-1n-1a1 are distinct. An abelian group is R*-sequenceable if it has an R-sequencing a1,a2,...,an-1 such that ai-1ai+1=ai for some i (subscripts are read modulo n-1). Friedlander, Gordon and Miller (1978) showed that an R*-sequenceable Sylow 2-subgroup is a sufficient condition for a group to be R-sequenceable. In this paper we also show that all noncyclic abelian 2-groups are R*-sequenceable except for 2 x 4 and 2 x 2 x 2.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/31392/1/0000306.pd

Hamiltonian decomposition of K∗n, patterns with distinct differences, and Tuscan squares

AbstractThis paper presents a few constructions for the decomposition of the complete directed graph on n vertices into n Hamiltonian paths. Some of the constructions will apply for even n and others to odd n. The constructions will be obtained from some patterns with distinct differences. The constructions will be exhibited by squares (called Tuscan squares) which sometimes are Latin squares (called Roman squares), and sometimes are not Latin. These squares have some special properties which are discussed in this paper