Edge homogeneous embeddings of cycles in graphs

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