On arithmetic deficiency for bicliques

It is known that edge-magic-ness can be applied to the arrangement of devices of a wireless network, which is a special case of a general concept of arithmetic edge-antimagicness. A graph G with p vertices and q edges is called (a, d)-edge-antimagic if there exists an injective vertex labeling function f: V(G) → {1, 2, ⋯, p} such that the induced edge labels, which are defined by f(uv) = f(u) + f(v) for each uv ∈ E(G), form an arithmetic progression {a, a + d, a + 2d, ⋯⋯, a + (q - 1)d} where d is a positive integer. The (a, d)-edge-antimagic deficiency μ
_d
(G) of a graph G, which is the least integer k such that G is (a, d)-edge-antimagic by modifying the range of the injective vertex labeling function from {1, 2, ⋯, p} to {1, 2, ⋯, p + k}. In this article, we completely determine the (a, d)-edge-antimagic deficiency of complete bipartite graphs K
_m,n
, which in particular confirms a conjecture raised by R. Figueroa-Centeno et al. when d = 1.