On Nice (1,1) Edge-Magic Graphs

By a (1, 1) edge-magic labeling of a (p, q) graph G we mean a bijection f : V ( G ) ∪ E ( G ) → { 1 , … , p + q } such that f ( u ) + f ( v ) + f ( u v ) = k is a constant for any edge uv of G. We call a graph G (1, 1) edge-magic if it has a (1, 1) edge-magic labeling f and in which case, the integer k is called the common edge count of f. We further call f a nice (1, 1) edge-magic labeling of G if f ( V ( G ) ) = { 1 , … , p } . The corresponding G is called a nice (1, 1) edge-magic graph. In this paper, we obtain a necessary and sufficient condition for a graph to be nice (1, 1) edge-magic and establish the niceness of several families of graphs. We also obtain a few general results. Also we investigate the relationship of a nice (1, 1) edge-magic labeling with other additive labelings like sequential, harmonious, α-valuation, cordial labeling etc. Some open problems are also raised.