Zk-Magic Labeling of Cycle of Graphs (original) (raw)
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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.
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.
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.
Cubo (Temuco), 2019
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+ definedas f +(v) = Pf(uv) taken over all edges uv incident at v is a constant. An A-magic graph G is said to be Zk-magic graph if the group A is Zk, the group of integers modulo k and these graphs are referred as k-magic graphs. In this paper we prove that the graphs such as path union of cycle, generalized Petersen graph, shell, wheel, closed helm, double wheel, flower, cylinder, total graph of a path, lotus inside a circle and n-pan graph are Zk-magic graphs.
International Journal of Mathematics And its Applications E k-Super Vertex Magic Labeling of Graphs
2018
Let G be a graph with p vertices and q edges. An Ek-super vertex magic labeling (Ek-SVML) is a bijection f : V (G) ∪ E(G) → {1, 2, . . . , p + q} with the property that f(E(G)) = {1, 2, . . . , q} and for each v ∈ V (G), f(v) + wk(v) = M for some positive integer M . For an integer k ≥ 1 and for v ∈ V (G), let wk(v) = ∑ e∈Ek(v) f(e), where Ek(v) is the set of all edges which are at distance at most k from v. The graph G is said to be Ek-regular with regularity r if and only if |Ek(e)| = r for some integer r ≥ 1 and for all e ∈ E(G). A graph that admits an Ek-SVML is called Ek-super vertex magic (Ek-SVM). This paper contain several properties of Ek-SVML in graphs. A necessary and sufficient condition for the existence of Ek-SVML in graphs has been obtained. Also, the magic constant for Ek-regular graphs has been obtained. Further, we establish E2-SVML of some classes of graphs such as cycles, complement of cycles, prism graphs and a family of circulant graphs. MSC: 05C78.
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.