Necessary and Sufficient Conditions for Unit Graphs to Be Hamiltonian

A graph is Hamiltonian if it has a cycle that visits every vertex exactly once; such a cycle is called a Hamiltonian cycle. In general, the problem of finding a Hamiltonian cycle in a given graph is an NP-complete problem and a special case of the traveling salesman problem. It is a problem in combinatorial optimization studied in operations research and theoretical computer science; see [Garey and Johnson 1979]. The only known way to determine whether a given graph has a Hamiltonian cycle is to undertake an exhaustive search, and until now no theorem giving a necessary and sufficient condition for a graph to be Hamiltonian was known. The study of Hamiltonian graphs has long been an important topic. See [Gould 2003] for a survey, updating earlier surveys in this area. Let n be a positive integer, and let Zn be the ring of integers modulo n. Grimaldi [1990] defined a graph G(Zn) based on the elements and units of Zn . The vertices of G(Zn) are the elements of Zn , and distinct vertices x and y are defined to be adjacent if and only if x + y is a unit of Zn . For a positive integer m, it follows that G(Z2m) is a φ(2m)-regular graph, where φ is the Euler phi function. In case m ≥ 2, the graph G(Z2m) can be expressed as the union of φ(2m)/2 Hamiltonian cycles. The odd case is not quite so easy, but the structure is clear and the results are similar to the even case. We recall that a cone over a graph is obtained by taking