Homogeneous embeddings of cycles in graphs

If G and H are vertex-transitive graphs, then the framing number fr(G, H) of G and H is defined as the minimum order of a graph every vertex of which belongs to an induced G and an induced H. This paper investigates fr(C-m, C-n) for m < n. We show first that fr(C-m, C-n) greater than or equal to n + 2 and determine when equality occurs. Thereafter we establish general lower and upper bounds which show that fr(C-m, C-n) is approximately the minimum of n - m + 2 root n and n + n/m.