On the k-edge magic graphs (original) (raw)
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Journal of Graph Labeling,, 2018
For any non-trivial abelian group A under addition a graph G is said to be A-magic if there exists a labeling f : E(G) → A − {0} such that, the vertex labeling f + defined as f + (v) = ∑ f (uv) taken over all edges uv incident at v is a constant. An A-magic graph G is said to be Z k-magic graph if the group A is Z k the group of integers modulo k. These Z k-magic graphs are referred as k-magic graphs. In this paper we prove that the total graph, square graph, splitting graph, middle graph, m∆ n-snake graph are Z k-magic graphs. Furthermore, let C be any cycle of even length n. Let G be the graph obtained by joining the vertices u i and u i+1 by a path of length m i for 1 ≤ i ≤ n − 1, and u 1 and u n by a path of length m n. Then G is Z k-magic if either all m 1 , m 2. .. m n are even or all are odd.
A generalization of magic labeling of two classes of graphs
Bull. ICA, 2019
A k-magic labeling of a finite, simple graph with |V (G)| = p and |E(G)| = q, is a bijection from the set of edges into the set of integers {1, 2, 3, • • • , q} such that the vertex set V can be partitioned into k sets V 1 , V 2 , V 3 , • • • , V k , 1 ≤ k ≤ p, and each vertex in the set V i has the same vertex sum and any two vertices in different sets have different vertex sum, where a vertex sum is the sum of the labels of all edges incident with that vertex. A graph is called k-magic if it has a k-magic labeling. The study of k-magic labeling is very interesting, since all magic graphs are 1-magic and all antimagic graphs are p-magic. The Splendour Spectrum of a graph G, denoted by SSP (G), is defined by SSP (G) = {k | G has a k-magic labeling}. In this paper, we determine SSP (K m,n), m and n are even and SSP (T n), where T n is the friendship graph and n ≥ 1.
Journal of Algorithms and Computation, 2018
For any non-trivial abelian group A under addition a graph G is said to be A-magic if there exists a labeling f : E(G) → A − {0} such that, the vertex labeling f + defined as f + (v) = f (uv) taken over all edges uv incident at v is a constant. An A-magic graph G is said to be Z k-magic graph if the group A is Z k the group of integers modulo k. These Z k-magic graphs are referred to as k-magic graphs. In this paper we prove that the total graph, flower graph, generalized prism graph, closed helm graph, lotus inside a circle graph, G ⊙ K m , m-splitting graph of a path and m-shadow graph of a path are Z k-magic graphs.
On New Properties of Graphs with Magic Type Labeling
Control Systems and Computers, 2019
We have shown the connection between vertex labelings of magic graph and its overgraph. Also, we have introduced binary relation on the set of all D i-distance magic graphs, where D i ⊂ {0, 1, ..., d}, i = 1, 2, ... and proved, that it is equivalence relation. Therefore, we have explored the properties of the graphs, which are in this relation. Finally, we have offered the algorithm of constructing r-regular handicap graph G = (V, E) of order n, where n ≡ 0(mod8), r ≡ 1,3(mod4) and 3 ≤ r ≤ n-5.
On 0-Edge Magic Labeling of Some Graphs Research
2017
A graph G = (V,E) where V = {vi, 1 ≤ i ≤ n} and E = {vivi+1, 1 ≤ i ≤ n} is 0-edge magic if there exists a bijection f : V (G)→ {1,−1} then the induced edge labeling f : E → {0}, such that for all uv ∈ E(G), f∗(uv) = f(u) + f(v) = 0. A graph G is called 0-edge magic if there exists a 0-edge magic labeling of G. In this paper, we determine the 0-edge magic labeling of the cartesian graphs Pm × Pn and Cm × Cn, and the generalized Petersen graph P (m,n). MSC: 05C78.
On a question by Erdős about edge-magic graphs
Discrete Mathematics, 1999
A (p; q) graph is edge-magic if the vertices and edges can be labeled with distinct elements from the set 1; 2; : : : ; p + q in such a way that the sum is the same along any edge. We give some general results about edge-magic graphs and show precisely which complete graphs are edge-magic.
Vertex Bi-magic Graphs from Magic and Anti-magic Graphs
2016
Let G(V,E) be a graph of order p and sizeq and let λ : V ∪ E → {1, 2, ..., p + q} be a bijective mapping. λ is called a vertex magic total labeling ofG if at each vertex x, the vertex weight under this λ, wtλ(x) = λ(x) + ∑ xy∈E λ(xy) = α, a constant. λ is called a vertex bi-magic total labeling ofG if the vertex weight at each vertex is eitherα or β, where α, β are two fixed constants. It is called (a, d) vertex anti-magic total labeling of G if the set of vertex weights of all vertices inG is {a, a+d, a+2d, ..., a+(p−1)d}, where a, d > 0 are integers. In this article, we introduce two other variations of bi-magic labeling namely (1, 0) vertex bi-magic and (0, 1) vertex bi-magic and also discuss new techniques of generating (1, 1) vertex bi-magic, (1, 0) vertex bi-magic and (0, 1) vertex bi-magic graphs using some operations on vertex magic and vertex anti-magic graphs.
On Complementary Edge Magic Labeling ofCertain Graphs
American Journal of Mathematics and Statistics, 2012
By G(p, q) we denote a graph having p vertices and q edges, by V(G) and E(G) the vertex set and the edgeset of G respectively. But the vertices and edges are called the elements of the graph. A (p, q)-graph G is called the edgemagic if there exists a bijective function f: V(G) U E(G) {1,2, ,p+q} such that f(u)+f(v)+f(uv)=k is a constant called the valence of f for any edge uv of G. Given an edge magic f of a graph G(p, q) the function () f x such that () f x =p+q+1-f(x) for all elements of G is said to be complementary to f(x) or complementary edge magic labeling () f x. The purpose of this article is to search for certain graphs Km, n (m, n ≥ 1), Cn (n ≥ 3), np2, f n (fan) Bn (bwk) and nG (n ≥ 2) where G is bipartite or tripartite which have complementary edge magic strength.
E-super vertex magic labelings of graphs
Discrete Applied Mathematics, 2012
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 is E-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 H m,n and on some necessary conditions for a graph to be E-super vertex magic.
On the construction of super edge-magic total graphs
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.