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, . . . , 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)-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)-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 - 1 and maximum degree less than k + 2, and we exhibit an infinite family of bracelet graphs with minimum degree k - 1 and maximum degree k ...

• Type: Journal
• Year: 2008
• Where: DM
• Authors: Karola Mészáros
• Keywords: Distinct Vertices, DM 2008, Hamiltonian Cycle, Simple Graph