Hamilton paths in generalized Petersen graphs

Abstract

Alspach and Qin proved that connected Cayley graphs of Hamiltonian groups (all subgroups are normal) are either Hamilton-connected (every pair of vertices is joined by a Hamilton path), or are bipartite and Hamilton-laceable (every pair on opposite sides of the bipartition are joined by a Hamilton path). Their proof made use of Hamilton-connectedness of certain Generalized Petersen graphs.In this work, we extend (and make a small correction to) the results of Alspach and Liu on Hamilton paths in generalized Petersen graphs. Alspach and Liu showed that, for k∈{1,2,3} and gcd(n,k)=1, P(n,k) is either Hamilton-connected or bipartite and Hamilton-laceable, as long as (n,k)≠(6r+5,2) or (5,3). For k=2, we consider the remaining cases for n and completely determine which pairs of vertices in P(n,2) are joined by Hamilton paths. However, the main point is to show that, for each k, it is a finite problem to determine, for all n, which pairs of vertices in P(n,k) are the ends of a Hamilton path.