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

TL;DR

The paper investigates odd $n$ for which the complete graph $K_n$ has an orthogonal double cover (ODC) by Hamiltonian paths. It proves that Anderson and Preece's discrete-logarithm terrace construction for $\\$Z_{2n+1}$, when $2n+1$ is prime, is an ODC-starter, enabling translates to form an ODC of $K_n$. This yields infinitely many new $n$ values (via primes $p \equiv 7 \pmod{8}$ giving $n=(p-1)/2$ and via Sophie Germain primes) and provides explicit constructions, including an example at $n=15$. By linking discrete-log terraces to ODC-starters, the work broadens the known catalog of $n$ for which $K_n$ admits an ODC by Hamiltonian paths and suggests new infinite families for combinatorial designs.

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.

Figures (1)

• Figure 1: An orthogonal double cover of $K_9$ by Hamiltonian paths.

Theorems & Definitions (10)

• Theorem 1.1
• proof : Proof note
• Theorem 1.2
• Lemma 2.1
• Example 2.2
• Theorem 2.3
• proof
• Theorem 2.4
• proof
• Example 2.5