Generalizing Pancyclic and k-Ordered Graphs

Given positive integers k m n, a graphG of order n is ðk;mÞ-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 ðk;mÞ-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 ðk;mÞ-pancyclic ordered. Minimum degree conditions and minimum sum of degree conditions for nonadjacent vertices that imply a graph is ðk;mÞ-pancylic ordered are also proved. Examples showing that these constraints are best possible are provided.