Frontiers of Mathematics in China >
Super (a, d)-edge-antimagic total labelings of complete bipartite graphs
Received date: 25 Jul 2017
Accepted date: 10 Oct 2017
Published date: 12 Jan 2018
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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)-edgeantimagic 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.
Key words: Graph Km,n; ; super (a; d)-edge-antimagic total labeling; matrix
Zhihe LIANG . Super (a, d)-edge-antimagic total labelings of complete bipartite graphs[J]. Frontiers of Mathematics in China, 2018 , 13(1) : 129 -146 . DOI: 10.1007/s11464-017-0671-y
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