Distance magic (r, t)-hypercycles

Abstract

A hypergraph H is a pair H - (V,E) where V is a set of vertices and E is a set of non-empty subsets of V called hyperedges. If all edges have the same cardinality t, the hypergraph is said to be t'-uniform. Let H = (V, E) be a hypergraph of order n. A distance magic labeling of H is a bijection l: V → {1,2,..., n} for that there exists a positive integer k such that Σ_x∈N(v) l(x) = k for all v ∈ V, where N(v) is the open neighborhood of v. The (r, t)-hypercycle, 1 ≤ r ≤ t - 1, is defined as t-uniforin hypergraph whose vertices can be ordered cyclically in such a way that the edges are segments of that cyclic order and every two consecutive edges share exactly r vertices. It was proved that (1,2)-hypercycle of order n is a distance magic graph if and only if n = 4 ([5]). In this paper we prove that if p is odd, then the p-th power of a cycle C_n is a distance magic graph if and only if 2p(p + 1) = 0 (mod n), n ≥ 2p + 2 and n/gcd(n,p+1) ≡ 0 (mod 2). Using this fact we we solve the problem of distance magic (r, t)-hypercycles for t = 3,4,6.