Constructions of hamiltonian graphs with bounded degree and diameter O (log n)

Token ring topology has been frequently used in the design of distributed loop computer networks and one measure of its performance is the diameter. We propose an algorithm for constructing hamiltonian graphs with $n$ vertices and maximum degree $\Delta$ and diameter $O (\log n)$, where $n$ is an arbitrary number. The number of edges is asymptotically bounded by $(2 - \frac{1}{\Delta - 1} - \frac{(\Delta - 2)^2}{(\Delta - 1)^3}) n$. In particular, we construct a family of hamiltonian graphs with diameter at most $2 \lfloor \log_2 n \rfloor$, maximum degree 3 and at most $1+11n/8$ edges.

💡 Research Summary

Abstract and Motivation
Token‑ring topologies are widely used in distributed loop networks because they provide a simple, deterministic way for nodes to share a common communication medium. In such systems the diameter—the longest shortest‑path distance between any two nodes—directly influences latency and scalability. While many studies have examined degree‑diameter trade‑offs, far fewer have addressed the additional requirement that the network contain a Hamiltonian cycle (a closed walk visiting every node exactly once). This requirement is essential for token‑ring operation, yet it severely restricts the class of admissible graphs. The present paper tackles the problem of constructing, for any integer n, a Hamiltonian graph with maximum degree Δ, diameter O(log n), and a provably small number of edges.

Main Contributions

1. An O(n)‑time deterministic algorithm that, given n and a degree bound Δ ≥ 3, outputs a Hamiltonian graph whose diameter does not exceed a constant multiple of log₂ n.

2. A tight asymptotic bound on the total number of edges:

   \