Super (a, d)-edge-antimagic total labelings of complete bipartite graphs

Abstract

An (a, d)-edge-antimagic total labeling of a graph G is a bijection f from V(G) ∪ E(G) onto {1, 2,…,|V(G)| + |E(G)|} with the property that the edge-weight set {f(x) + f(xy) + f(y) | xy ∈ E(G)} is equal to {a, a + d, a + 2d,...,a + (|E(G)| − 1)d} for two integers a > 0 and d ⩾ 0. An (a, d)-edge-antimagic total labeling is called super if the smallest possible labels appear on the vertices. In this paper, we completely settle the problem of the super (a, d)-edge-antimagic total labeling of the complete bipartite graph Km,n and obtain the following results: the graph Km,n has a super (a, d)-edge-antimagic total labeling if and only if either (i) m = 1, n = 1, and d ⩾ 0, or (ii) m = 1, n ⩾ 2 (or n = 1 and m ⩾ 2), and d ∈ {0, 1, 2}, or (iii) m = 1, n = 2 (or n = 1 and m = 2), and d = 3, or (iv) m, n ⩾ 2, and d = 1.