Minimum Degree Conditions for Cycles Including Specified Sets of Vertices

This paper generalises the concept of vertex pancyclic graphs. We define a graph as set-pancyclic if for every set S of vertices there is a cycle of every possible length containing S. We show that if the minimum degree of a graph exceeds half its order then the graph is set-pancyclic. We define a graph as k-ordered-pancyclic if, for every set S of cardinality k and every cyclic ordering of S, there is for every possible length a cycle of that length containing S and encountering S in the specified order. We determine the best possible minimum-degree condition which guarantees that a graph is k-ordered-pancyclic.