On (a,1)-vertex-antimagic edge labeling of regular graphs

An (a,s)-vertex-antimagic edge labeling (or an (a,s)-vae labeling,for short) of g is a bijective mapping from the edge set e(g) of a graph g to the set of integers 1,2,.,|e(g)| with the property that the vertex-weights form an arithmetic sequence starting from a and having common difference s,where a and s are two positive integers,and the vertex-weight is the sum of the labels of all edges incident to the vertex. a graph is called (a,s)-antimagic if it admits an (a,s)-vae labeling. in this paper,we investigate the existence of (a,1)-vae labeling for disconnected 3-regular graphs. also,we define and study a new concept (a,s)-vertex-antimagic edge deficiency,as an extension of (a,s)-vae labeling,for measuring how close a graph is away from being an (a,s)-antimagic graph. furthermore,the (a,1)-vae deficiency of hamiltonian regular graphs of even degree is completely determined. more open problems are mentioned in the concluding remarks. © 2015 martin bača et al.