New Constructions of Antimagic Graph Labeling
نویسندگان
چکیده
An anti-magic labeling of a finite simple undirected graph with p vertices and q edges is a bijection from the set of edges to the set of integers {1, 2, ..., q} such that the vertex sums are pairwise distinct, where the vertex sum at vertex u is the sum of labels of all edges incident to such vertex. A graph is called anti-magic if it admits an antimagic labeling. Hartsfield and Ringel conjectured in 1990 that all connected graphs except K2 are anti-magic. Recently, N. Alon et al (Dense graphs are anti-magic, Journal of Graph Theory, Volume 47, Issue 4, (2004), pp. 297-309) showed that this conjecture is true for dense graphs, i.e. it is true for p-vertex graphs with minimum degree Ω(log p). More recently, D. Hefetz (Anti-magic graphs via the combinatorial nullstellensatz, Journal of Graph Theory, Volume 50, Issue 4, (2005), pp. 263-272) proved that in particular for a positive integer k, a graph G with 3 vertices is anti-magic if it admits a K3-factor. Tao-Ming Wang (Toroidal grids are anti-magic, COCOON 2005, Lecture Notes in Computer Science (LNCS) 3595 (2005), pp. 671-679) showed that higher dimensional torus graphs are antimagic. However the HartsfieldRingel conjecture is still unsettled while the research along this direction is extended. In this article, some new classes of sparse antimagic graphs are constructed through Cartesian products.
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