The Hamiltonian property of the consecutive-3 digraph

Abstract

A consecutive-d digraph is a digraph G(d, n, q, r) whose n nodes are labeled by the residues module n and a link from node i to node j exists if and only if j = qi + k (mod n) for some Ic with r less than or equal to k less than or equal to r + d - 1. Consecutive-d digraphs are used as models for many computer networks and multiprocessor systems, in which the existence of a Hamiltonian circuit is important. Conditions for a consecutive-d graph to have a Hamiltonian circuit were known except for gcd(la, d) = 1 and d = 3 Or 4. It was conjectured by Du, Hsu, and Hwang that a consecutive-3 digraph is Hamiltonian. This paper produces several infinite classes of consecutive-3 digraphs which are not (respectively, are) Hamiltonian, thus suggesting that the conjecture needs modification.