Minimal Degree and (k, m)-Pancyclic Ordered Graphs

Abstract

Given positive integers k ≤ m ≤ n, a graph G of order n is (k, m)-pancyclic ordered if for any set of k vertices of G and any integer r with m ≤ r ≤ n, there is a cycle of length r encountering the k vertices in a specified order. Minimum degree conditions that imply a graph of sufficiently large order n is (k, m)-pancylic ordered are proved. Examples showing that these constraints are best possible are also provided.