Minimal chordal sense of direction and circulant graphs

TL;DR

This paper characterizes a specific type of sense of direction called minimal chordal sense of direction (CSD), proves that graphs admitting such a CSD are Hamiltonian, and shows they are equivalent to circulant graphs, providing an efficient recognition method.

Contribution

It identifies the class of k-regular graphs with a minimal CSD and establishes their equivalence to circulant graphs, including Hamiltonian properties and recognition algorithms.

Findings

Connected graphs with minimal CSD are Hamiltonian.

Minimal CSD graphs are equivalent to circulant graphs.

Recognition of these graphs is polynomial-time for fixed degree k.

Abstract

A sense of direction is an edge labeling on graphs that follows a globally consistent scheme and is known to considerably reduce the complexity of several distributed problems. In this paper, we study a particular instance of sense of direction, called a chordal sense of direction (CSD). In special, we identify the class of k-regular graphs that admit a CSD with exactly k labels (a minimal CSD). We prove that connected graphs in this class are Hamiltonian and that the class is equivalent to that of circulant graphs, presenting an efficient (polynomial-time) way of recognizing it when the graphs' degree k is fixed.