Generalizing Pancyclic and k-Ordered Graphs

Abstract.

Given positive integers k≤m≤n, a graph G of order n is (k,m)-pancyclic if for any set of k vertices of G and any integer r with m≤r≤n, there is a cycle of length r containing the k vertices. Minimum degree conditions and minimum sum of degree conditions of nonadjacent vertices that imply a graph is (k,m)-pancylic are proved. If the additional property that the k vertices must appear on the cycle in a specified order is required, then the graph is said to be (k,m)-pancyclic ordered. Minimum degree conditions and minimum sum of degree conditions for nonadjacent vertices that imply a graph is (k,m)-pancylic ordered are also proved. Examples showing that these constraints are best possible are provided.