A Discrete Logarithm Construction for Orthogonal Double Covers of the Complete Graph by Hamiltonian Paths

TL;DR

This paper demonstrates how a discrete logarithm-based construction for cyclic groups yields new orthogonal double covers of complete graphs by Hamiltonian paths, expanding known cases.

Contribution

It introduces a novel method linking discrete logarithms to orthogonal double covers, providing infinitely many new examples for complete graphs.

Findings

Constructs orthogonal double covers for infinitely many complete graphs

Uses discrete logarithm modulo a prime to build Hamiltonian path arrangements

Expands the set of known orthogonal double covers in graph theory

Abstract

During their investigation of power-sequence terraces, Anderson and Preece briefly mention a construction of a terrace for the cyclic group $\mathbb{Z}_n$ when $n$ is odd and $2n+1$ is prime; it is built using the discrete logarithm modulo $2n+1$. In this short note we see that this terrace gives rise to an orthogonal double cover (ODC) for the complete graph $K_n$ by Hamiltonian paths. This gives infinitely many new values for which such an ODC is known.