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On super \$(a,1)\$-edge-antimagic total labelings of regular graphs. (English)
Discrete Math. 310, No. 9, 1408-1412 (2010).
Summary: A labeling of a graph is a mapping that carries some set of graph elements into numbers (usually positive integers). An \$(a,d)\$-edge-antimagic total labeling of a graph with \$p\$ vertices and \$q\$ edges is a one-to-one mapping that takes the vertices and edges onto the integers \$1, 2\cdots ,p+q\$, so that the sum of the labels on the edges and the labels of their end vertices forms an arithmetic progression starting at \$a\$ and having difference \$d\$. Such a labeling is called super if the \$p\$ smallest possible labels appear at the vertices. In this paper we prove that every even regular graph and every odd regular graph with a 1-factor are super \$(a,1)\$-edge-antimagic total. We also introduce some constructions of non-regular super \$(a,1)\$-edge-antimagic total graphs.