E-Super Vertex Magic Regular Graphs of Odd Degree

2014

A vertex-magic labeling is an assignment of the integers from 1,2,3,.......m+n to the vertices and edges of G so that at each vertex , the vertex label and the labels on the edges incident at that vertex add to a fixed constant. In this paper, we introduce a new concept odd s vertex labeling of a graph an f establish some families of a graphs have odd super vertex magic labeling 1.INTRODUCTION In this paper, we consider only finite simple undirected graph. The graph G has vertex set V=V (G) and edge set E=E (G) and we take m=|E| and n =|V|. The set of vertices adjacent to a vertex u of G is denoted by N(u). In the notion of a vertex-magic total labeling was introduced. This is an assignment of the integers from 1 to m+n to the vertices and edges of G so that at each vertex, the vertex label and the labels on the edges incident at that vertex, add to a fixed constant more formally, a one to one map f from V ∪ E onto the integers {1,2,3.....m+n} is a vertex-magic total labeling if the...

Electronic Journal of Graph Theory and Applications

Suppose G = (V, E) be a simple graph with p vertices and q edges. An edge-magic total labeling of G is a bijection f : V ∪ E → {1, 2,. .. , p + q} where there exists a constant r for every edge xy in G such that f (x) + f (y) + f (xy) = r. An edge-magic total labeling f is called a super edge-magic total labeling if for every vertex v ∈ V (G), f (v) ≤ p. The super edge-magic total graph is a graph which admits a super edge-magic total labeling. In this paper, we consider some families of super edge-magic total graph G. We construct several graphs from G by adding some vertices and edges such that the new graphs are also super edge-magic total graphs.

Malaya Journal of Matematik

Let G be a simple graph with p vertices and q edges. A V-super vertex magic labeling is a bijection f : V (G) ∪ E(G) → {1, 2,. .. , p + q} such that f (V (G)) = {1, 2,. .. , p} and for each vertex v ∈ V (G), f (v) + ∑ u∈N(v) f (uv) = M for some positive integer M. A V k-super vertex magic labeling (V k-SVML) is a bijection f : V (G) ∪ E(G) → {1, 2,. .. , p + q} with the property that f (V (G)) = {1, 2,. .. , p} and for each v ∈ V (G), f (v) + w k (v) = M for some positive integer M. A graph that admits a V k-SVML is called V k-super vertex magic. This paper contains several properties of V k-SVML in graphs. A necessary and sufficient condition for the existence of V k-SVML in graphs has been obtained. Also, the magic constant for E k-regular graphs has been obtained. Further, we study some classes of graphs such as cycles, complement of cycles, prism graphs and a family of circulant graphs which admit V 2-SVML.

Lecture Notes in Computer Science, 2012

An antimagic labeling of a finite simple undirected graph with q edges is a bijection from the set of edges to the set of integers {1, 2, • • • , q} such that the vertex sums are pairwise distinct, where the vertex sum at vertex u is the sum of labels of all edges incident to such vertex. A graph is called antimagic if it admits an antimagic labeling. It was conjectured by N. Hartsfield and G. Ringel in 1990 that all connected graphs besides K2 are antimagic. Another weaker version of the conjecture is every regular graph is antimagic except K2. Both conjectures remain unsettled so far. In this article, certain classes of regular graphs of odd degree with particular type of perfect matchings are shown to be antimagic. As a byproduct, all generalized Petersen graphs and some subclass of Cayley graphs of Zn are antimagic.

Advances in theoretical and applied mathematics, 2016

A graph that admits a super edge-magic total labeling is called a SEMT graph. In this paper, we study the structure of connected unicyclic SEMT graphs with magic constant 3 =   q p . We also enumerate SEMT labelings with 3 =   q p  , for connected unicyclic graphs that contain a star of maximum size m, with 1 4     p m p

Malaya Journal of Matematik

Let G be a finite and simple (p, q) graph. An one-one onto function f : V (G) ∪ E(G) → {1, 2, 3,. .. , p + q} is called Vsuper vertex magic graceful labeling if f (V (G)) = {1, 2, 3,. .. , p} and for any vertex v ∈ V (G), ∑ u∈N(v) f (uv)− f (v) = M, where M is a whole number. For an integer k ≥ 1, let E k (v) = {e ∈ E(G) : the distance between e from v is less than or equal to k}. For v ∈ V (G), we define w k (v) = ∑ e∈E k (v) f (e). A V k-super vertex magic graceful labeling (V k-SVMGL) is a one-one function f from V (G) ∪ E(G) onto the set {1, 2, 3,. .. , p + q} such that f (V (G)) = {1, 2, 3,. .. , p} and for any element v ∈ V (G), we have w k (v) − f (v) = M, where M is a whole number. In this paper, we study several properties of V k-SVMGL and we identify an equivalent condition for the E k-regular graphs which admits V k-SVMGL. At last we identify some families of graphs which admit V 2-SVMGL.

Malaya Journal of Matematik, 2019

An H-magic labeling in an H-decomposable graph G is a bijection f : V (G) ∪ E(G) → {1, 2,. .. , p + q} such that for every copy H in the decomposition, ∑ v∈V (H) f (v) + ∑ e∈E(H) f (e) is constant. The function f is said to be H-V-super magic labeling if f (V (G)) = {1, 2,. .. , p}. In this article, we give a few fundamental properties of H-V-super magic labeling. Obtained the magic constant for H-factorable graphs which are H-V-super magic. Further we gave a necessary and sufficient condition for an even regular graph to be 2-factor-V-super magic.

Journal of Applied Mathematics, 2015

An(a,s)-vertex-antimagic edge labeling(or an(a,s)-VAElabeling, for short) ofGis a bijective mapping from the edge setE(G)of a graphGto the set of integers1,2,…,|E(G)|with the property that the vertex-weights form an arithmetic sequence starting fromaand having common differences, whereaandsare 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)-VAElabeling. 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.

Let G be a finite simple graph with v vertices and e edges. A vertex-magic total labeling is a bijection We study some of the basic properties of such labelings, find some families of graphs that admit super vertex-magic labelings and show that some other families of graphs do not.

Discrete Mathematics, 2007

Let G = (V , E) be a finite non-empty graph, where V and E are the sets of vertices and edges of G, respectively, and |V | = n and |E| = e. A vertex magic total labeling is a bijection from V ∪ E to the consecutive integers 1, 2,. .. , n + e with the property that for every v ∈ V , (v) + w∈N(v) (v, w) = h, for some constant h. Such a labeling is super if (V (G)) = {1, 2,. .. , n}. MacDougall, Miller and Sugeng proposed the conjecture: If n ≡ 0(mod 4), n > 4, then K n has a super vertex-magic total labeling (VMTL). We prove that this conjecture holds true by means of giving a family of super VMTLs of K 4l , l > 1.