Vertex-magic Labeling Research Articles

Group vertex magic labeling of some special graphs

For any additive abelian group $A$. Let $\mu$ be an element of $A$, a graph $G=(V,E)$ is said to be $A$-vertex magic graph if there exist a labeling function $f:V(G)\rightarrow A\setminus\{0\}$ such that $\omega(v)=\sum_{u\in N(v)} f(u)=\mu$ for any vertex $v$ of $G$, where $N(v)$ is the set of the open neighborhood of $v$. In this paper, we prove that the graphs such as wheel, Corona $C_{n}\odot mk$, subdivision of ladder and $t$-fold wheel for $t\neq n$ nor $n-2$ are $A$-vertex magic graphs. Also we prove that the subdivide wheel, helm and closed helm are $Z_{k}$-vertex magic graphs. However we prove that the triangular book and $t$-fold wheel for $t=n,n-2$ are group vertex magic graphs.

On Group-Vertex-Magic Labeling of Simple Graphs

Let A be an Abelian group with identity 0. The A-vertex-magic labeling of a graph G is a mapping from the set of vertices in G to A-{0} such that the sum of the labels of every open neighborhood vertex of v is equal, for every vertex v in G. In this article, we discuss group-vertex-magic labeling of some simple graphs by using the Abelian group Zk, with natural numbers k1. We investigated some classes of simple graphs are path graphs, complete graphs, cyclic graphs, and star graphs. The method we used in this article is literature study and then developing the properties of vertex-magic labeling of some simple graphs, that are path graphs, complete graphs, cyclic graphs, and star graphs. We obtain that complete graphs, cyclic graphs, and star graphs have Zk-vertex-magic labeling, while path graphs have vertex-magic labeling only for n=2,3.

Results on Hefty V4-Vertex Magic Graph Labeling

Let V4 be an abelian group under multiplication. Let g : E(G) →V4 −{1}. The vertex magic labeling on V4 is defined as the vertex labeling g∗ : V (G) →V4 such that ), where the product is taken over all edges uv of G incident at v is a constant. If the constant is 1, it becomes a Hefty V4-vertex magic labeling. A graph is said to be Hefty V4-magic graph if it admits a Hefty V4-vertex magic labeling. In this paper we investigate the Hefty V4-vertex magic labeling behaviour of graphs like Bipartite, Complete graph, Cayley graph etc.

On the products of group vertex magic graphs

Let be a simple undirected graph and let A be an additive Abelian group with identity 0. A mapping is said to be a A-vertex magic labeling of G if there exists a μ in A such that for any vertex v of G. If G admits such a labeling, then it is called an A-vertex magic graph. If G is A-vertex magic for any non-trivial Abelian group A, then G is called a group vertex magic graph. In this paper, we consider A-vertex magic and group vertex magic labeling of different products of graphs.

Consecutive z-index vertex magic labeling graphs

The conception of magic labeling in fuzzy graphs elongates to fuzzy vertex magic labeling together with consecutive non-integer values in (0, 1] and the graph’s repercussion is named as fuzzy consecutive vertex magic labeling graphs (FCVM) along with the z-index. In this manuscript, we give some properties associated with FCVM labeling along with z-index as well as the presence of FCVM labeling with z-index in trees and some generalizations. Moreover, we examine the FCVM labeling along with z-index of both regular and irregular graphs. Finally, in real-time applications, we bestow an instance for fuzzy consecutive vertex magic labeling graphs.

Vertex Magic Labeling On V_4 for Cartesian product of two cycles

Let be an abelian group under multiplication. Let . Then the vertex magic labeling on is induced as such that where the product is taken over all edges of incident at is constant. A graph is said to be - magic if it admits a vertex magic labeling on . In this paper, we prove that , , Generalized fish graph, Double cone graph and four Leaf Clover graph are all -magic graphs.

E-super vertex magic labeling of cartoon flowers and wounded flowers

In this paper, we define a new class of graph called the cartoon flower and then show that E-super vertex magic labeling does not exist for the class of cartoon flower graphs. We further define the wounded cartoon flower graphs and establish some sufficient conditions for the graph not to be E-super vertex magic. We also give examples of some wounded cartoon flowers that admit an E-super vertex magic labeling and some others that do not.

Application of Reverse Vertex Magic Labeling of a Graph

Graph labeling is a currently emerging area in the research of graph theory. A graph labeling is an assignment of integers to the vertices or edges or both subject to certain conditions. If the labels of edges are distinct positive integers and for each vertex the sum of the labels of all edges incident with is the same for every vertex in the given graph, then the labeling of the graph is called magic labeling. There are several types of magic labeling defined on graphs. In this paper we consider vertex magic labeling and group magic labeling of graphs as an application of magic labeling. We solve a problem of finding number of computers/workstations to be allocated to each department in a company under certain conditions.

Vertex magic total labeling of middle and total graph of cycle

A vertex magic total labeling is a bijection from the union of the vertex set and edge set to the consecutive integers 1,2,3,....,v+e with the property that for every u in the vertex set, the sum of the label of u and the label of the edges incident with u is equal to k, for some constant k. In this paper, we establish the vertex magic labeling of some classes of graphs and provide some open problems related to it.

Mod(k) Vertex Magic Labeling of Some Hamiltonian Graphs

Mod(k) Vertex Magic Labeling of Some Hamiltonian Graphs

Mod(k) Vertex Magic Labeling in Generalized 2-complement of some Graphs- Paper II

Mod(k) Vertex Magic Labeling in Generalized 2-complement of some Graphs- Paper II

Reverse vertex magic labeling of complete graphs

A reverse vertex magic labeling of a graph G = (V,E) is a bijection such that for every vertex , is a constant . It is well known that all complete graphs Kn admit a reverse magic labeling. In this paper, we present a new proof of this theorem using the concepts of twin factorization and magic square.

On MOD(k) Vertex Magic Labeling of Graphs

A (p, q) graph G with p vertices and q edges is mod(k)vertex magic if for any integer k≥2, lɛZk and there exists a one-to-one map f from V(G) to and such that for each edge e, the sum of the labels of the vertices adjacent with e are all equal to the same constant modulo k. In this paper, we introduce a new type of labeling namely mod(k) vertex magic labeling of a graph and we prove some classes of graphs are mod(k)vertex magic.

Regular sparse anti-magic squares with small odd densities

Sparse anti-magic squares are useful in constructing vertex-magic labelings for bipartite graphs. An n×n array based on {0,1,…,nd} is called a sparse anti-magic square of ordernwith densityd (d<n), denoted by SAMS(n,d), if its row-sums, column-sums and two main diagonal sums constitute a set of 2n+2 consecutive integers. A SAMS(n,d) is called regular if there are d positive entries in each row, each column and each main diagonal. In this paper, we investigate the existence of regular sparse anti-magic squares with densities d=3,5 and it is proved that there exists a regular SAMS(n,3) if and only if n≥4 and there exists a regular SAMS(n,5) if and only if n≥6.

Solution to some open problems on E-super vertex magic labeling of disconnected graphs

An E-super vertex magic labeling is a bijection f :V(G)UE(G)→{1,2,3,…,p+q} such that for each vertex u, f(u)+∑v∈N(u)f(uv)=k for some constant k where f(E(G))={1,2,3,…,q}. A graph that admits an E-super vertex magic labeling is called an E-super vertex magic graph. The only disconnected graphs that have been shown to be E-super vertex magic are mCn if and only if both m and n are odd. The article “Marimuthu and Balakrishnan (2012)” discussed the E-super vertex magicness of connected graphs. In this paper, we pay our attention to prove the existence and non existence of E-super vertex magic labeling for some families of disconnected graphs. Also we provide solution to some open problems found in the article “Gray and MacDougall (2009)”.

On V-super and E-super vertex-magic total labelings of graphs

Let G be a graph of order p and size q. A vertex-magic total labeling is an assignment of the integers 1,2,…,p+q to the vertices and the 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, called the magic constant of G. Such a labeling is V-super vertex-magic total if f(V(G))={1,2,…,p} and is E-super vertex-magic total if f(E(G))={1,2,…,q}. A graph that admits a V-super vertex-magic total labeling is called V-super vertex-magic. Similarly, a graph that admits an E-super vertex-magic total labeling is called E-super vertex-magic. In this paper, we prove a necessary and sufficient condition for the existence of V-super vertex-magic labeling. Further, we find V-super vertex-magic total labeling and E-super vertex-magic total labeling of some families of graphs.

Note on [formula omitted]-super vertex magic graphs

Let G=(V(G),E(G)) be a finite simple graph with p=|V(G)| vertices and q=|E(G)| edges, without isolated vertices or isolated edges. A vertex magic total labeling is a bijection from V(G)∪E(G) to the consecutive integers 1,2,…,p+q, with the property that, for every vertex u in V(G), one has f(u)+∑uv∈E(G)f(uv)=k for some constant k. Such a labeling is called E-super vertex magic if f(E(G))={1,2,…,q}. A graph G is called E-super vertex magic if it admits an E-super vertex magic labeling. More recently Marimuthu and Balakrishnan (2012) studied some basic properties of such labeling and established E-super vertex magic labeling of some families of graphs. In this note we extend their results and more examples are also provided.

Regular sparse anti-magic squares with the second maximum density

Sparse anti-magic squares are useful in constructing vertex-magic labelings for bipartite graphs. An n×n array based on {0,1,⋯,nd} is called a sparse anti-magic square of order n with density d (d<n), denoted by SAMS(n,d), if its row-sums, column-sums and two main diagonal sums form a set of 2n+2 consecutive integers. An SAMS(n,d) is called regular if there are d positive entries in each row, each column and each main diagonal. In this paper, we investigate the existence of regular sparse anti-magic squares with the second maximum density, i.e., d=n−2, and it is shown that there exists a regular SAMS(n,n−2) if and only if n≥4.

Formula omitted]-super vertex magic labelings of graphs

Let G be a finite simple graph with p vertices and q edges. A vertex magic total labeling is a bijection from V(G)∪E(G) to the consecutive integers 1,2,3,…,p+q with the property that for every u∈V(G),f(u)+∑v∈N(u)f(uv)=k for some constant k. Such a labeling isE-super if f(E(G))={1,2,3,…,q}. A graph G is called E-super vertex magic if it admits a E-super vertex magic labeling. In this paper, we study some basic properties of such labelings and we establish E-super vertex magic labeling of some families of graphs. The main focus of this paper is on the E-super vertex magicness of Hm,n and on some necessary conditions for a graph to be E-super vertex magic.

Vertex-magic labeling of regular graphs: Disjoint unions and assemblages

We establish the existence of vertex-magic total labelings (VMTLs) for several infinite classes of regular graphs. The main method of construction is to assemble a number of appropriately labeled copies of one graph into a single graph with a VMTL. This method enables us for example to begin with any even-regular graph and from it construct a cubic graph possessing a VMTL. An important feature of the construction is that it produces strong VMTLs for many even order regular graphs. In addition the method provides another proof that for any odd-regular graph G possessing a VMTL, the disconnected graph tG has a VMTL for all t≥1. The construction also extends to certain families of non-regular graphs.