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 ∈ ...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 smMlest 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 [（m,n and obtain the following results： the graph t（m,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 （orn=1 and m≥2）,and d ∈{0,1,2},or （iii） m=l,n=2 （orn=1 and m = 2）, and d= 3, or （iv） m,n≥2, and d=1展开更多
文摘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 smMlest 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 [（m,n and obtain the following results： the graph t（m,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 （orn=1 and m≥2）,and d ∈{0,1,2},or （iii） m=l,n=2 （orn=1 and m = 2）, and d= 3, or （iv） m,n≥2, and d=1
