Edge homogeneous embeddings of cycles in graphs

Let n > m 2:: 4 be positive integers. The edge framing number efr( Cm, Cn) of Cm and Cn is defined as the minimum size of a graph every edge of which belongs to an induced Cm and an induced Cn. We show that efr(Cm,Cn) = n+4 ifn = 2m-4 and m 2:: 5, efr(Cm,Cn) = n+5 if n = 2m-6 and m ~ 7 and efr(Cm1 Cn) = n+6 if n = 2m-8 (m ~ 10) or m = n -1 (where n ~ 5 and n ¢ {6,S}) or m = n 2 (n = 6 or n 2:: 9). It is also shown that efr(Cm1 Cn) ~ n + 6 for n > m ~ 4 with n i=2m 4 or 2m 6 and (m,n) =f. (5,7). Furthermore, for the cases n = 2m 4 (m ~ 5) and n = 2m 6 (m ~ 7) we show that Cm and Cn are uniquely edge framed.