, Volume 28, Issue 9, pp 1865-1874
Date: 11 May 2012

Vertex-antimagic labelings of regular graphs


Let G = (V,E) be a finite, simple and undirected graph with p vertices and q edges. An (a, d)-vertex-antimagic total labeling of G is a bijection f from V (G) ∪ E(G) onto the set of consecutive integers 1, 2, …, p + q, such that the vertex-weights form an arithmetic progression with the initial term a and difference d, where the vertex-weight of x is the sum of the value f(x) assigned to the vertex x together with all values f(xy) assigned to edges xy incident to x. Such labeling is called super if the smallest possible labels appear on the vertices.

In this paper, we study the properties of such labelings and examine their existence for 2r-regular graphs when the difference d is 0, 1, …, r + 1.

Supported by Slovak VEGA Grant 1/0130/12, Higher Education Commission Pakistan (Grant No. HEC(FD)/2007/555) and by the Ministry of Education of the Czech Republic (Grant No. MSM6198910027)