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(uv)=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.