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

TL;DR

This paper presents an algorithm to construct Hamiltonian graphs with bounded degree and logarithmic diameter, optimizing network design for distributed systems.

Contribution

It introduces a novel method for constructing Hamiltonian graphs with specific degree and diameter constraints, including explicit bounds on edges and diameter.

Findings

Constructed Hamiltonian graphs with diameter O(log n)

Maximum degree of the graphs is bounded by Δ

Number of edges is asymptotically optimized

Abstract

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.