Complementary Edge-Magic Total Labelings of Cycle Graphs

In an edge-magic total labeling, the integers 1, 2,..., p+q are assigned in a bijective manner to the vertices and edges of a graph G(p, q) in such a way that the edge-sum f(u) + f(uv) + f(v) remains constant at k for each and every edge uv. For every edge-magic labeling f, its corresponding labeling f̄, which can be expressed as f̄(x) = p + q + 1 − f(x), is also edge-magic. This results in a detectable change in the magic constant. In this study, cycle graphs C_n are investigated, and it is shown that every cycle has an edge-magic labeling and that the complement of that labeling is likewise edge-magic. As a result, every cycle possesses a complementary edge-magic property. For the residue classes n ≡ 0 (mod 4) and n ≡ 2 (mod 4), we present explicit constructs for odd and even cycles. These constructs enhance the conventional cycle constructions while stressing a separate extra meaning. The constructions are demonstrated through the use of illustrative instances and a labeled figure, which also serves to validate the ensuing constant edge sums for both the original and complementary labeling structures.