On low degree k-ordered graphs

A simple graph G is k-ordered (respectively, k-ordered hamiltonian) if, for any sequence of k   distinct vertices v1,…,vkv1,…,vk of G, there exists a cycle (respectively, a hamiltonian cycle) in G containing these k vertices in the specified order. In 1997 Ng and Schultz introduced these concepts of cycle orderability, and motivated by the fact that k  -orderedness of a graph implies (k-1)(k-1)-connectivity, they posed the question of the existence of low degree k-ordered hamiltonian graphs. We construct an infinite family of graphs, which we call bracelet graphs  , that are (k-1)(k-1)-regular and are k-ordered hamiltonian for odd k. This result provides the best possible answer to the question of the existence of low degree k-ordered hamiltonian graphs for odd k. We further show that for even k, there exist no k  -ordered bracelet graphs with minimum degree k-1k-1 and maximum degree less than k+2k+2, and we exhibit an infinite family of bracelet graphs with minimum degree k-1k-1 and maximum degree k+2k+2 that are k-ordered for even k. A concept related to k-orderedness, namely that of k-edge-orderedness, is likewise strongly related to connectivity properties. We study this relation and give bounds on the connectivity necessary to imply k-(edge-)orderedness properties.