A class of self-orthogonal 2-sequencings

Abstract

Let 2K _ndenote the complete multigraph on n vertices in which each edge has multiplicity two. If 2K _ncan be partitioned into Hamiltonian paths such that any two distinct paths have exactly one edge in common, write 2K _n→ P _n. This paper considerably expands the set of known positive integers n such that 2K _n→ P _n. The solutions found have application to other similar problems. The basic idea is to consider an algebraic formulation of the problem in terms of 2-sequencings (terraces) with additional properties. Construction of these 2-sequencings gives a special type of solution for which very few examples have been known. The constructions detailed here hold eventually for certain classes of prime powers. For example, it is shown that there is a positive integer N such that if N < p ⁿ≡ 5 (mod 8) and 3 is not a fourth power residue of GF[p ⁿ], then the additive group of GF[p ⁿ] has a 2-sequencing of the required type—a self-orthogonal 2-sequencing. Some of the solutions admit a 2-coloring which is important for applications. The method of construction appears to be much better than the theoretical bounds that are obtained. The general bounds are found by means of a character sum argument.