Vertex magic total labeling of unions of generalized Petersen graphs and unions of special circulant graphs (original) (raw)
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Super vertex-magic total labelings of graphs
2004
Let G be a finite simple graph with v vertices and e edges. A vertex-magic total labeling is a bijection λ from V (G)∪E(G) to the consecutive integers 1, 2, • • • , v +e with the property that for every x ∈ V (G), λ(x) + Σ y∈N (x) λ(xy) = k for some constant k. Such a labeling is super if λ(V (G)) = {1, • • • , v}. 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.
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.
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.
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.
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 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.
2001
Various graph labelings that generalize the idea of a magic square have been discussed. In particular a magic labeling on a graph with v vertices and e edges will be defined as a one-to-one map taking the vertices and edges onto the integers 1, 2, ... , v+e with the property that the sum of the label on an edge and the labels of its endpoints is constant independent of the choice of edge. Properties of these labelings are surveyed and the question of which families of graphs have magic labelings are addressed. 1 Graph labelings All graphs in this paper are finite, simple and undirected (although the imposition of directions will cause no complications). The graph G has vertex-set V(G) and edge-set E(G). A general reference for graph-theoretic ideas is [19].
Degree-Magic Labelings on the Join and Composition of Complete Tripartite Graphs
Communications in Mathematics and Applications, 2019
A graph is called supermagic if there is a labeling of edges, where all edges are differently labeled with consecutive positive integers such that the sum of the labels of all edges, which are incident to each vertex of this graph, is a constant. A graph G is called degree-magic if all edges can be labeled by integers 1, 2,. .. , |E(G)| so that the sum of the labels of the edges which are incident to any vertex v is equal to (1 + |E(G)|) deg(v)/2. Degree-magic graphs extend supermagic regular graphs. In this paper, the necessary and sufficient conditions for the existence of degree-magic labelings of graphs obtained by taking the join and composition of complete tripartite graphs are found.
Sharp Bounds on Vertex N -magic Total Labeling Graphs
Mathematics and Statistics, 2024
A vertex N-magic total labeling is a bijective function that maps the vertices and edges of a graph G onto the successive integers from 1 to p + q. The labeling exhibits two distinct properties: First, the count of unique magic constants k i for i belonging to the set {1, 2, ..., N } is equivalent to the cardinality of N ; secondly, the magic constants k i must be arranged in a strictly ascending order. In the present context, the constant N is employed to represent different degrees of vertices. The term "magic constant values k i " for i ∈ {1, 2, ..., N } refers to specific numbers that exhibit unique and interesting properties and are employed in the context of this investigation. By adding up the weights of each vertex in V (G), we might receive a magical constant number k i for i ∈ {1, 2, ..., N }. Within the scope of this study, we discuss the sharp bounds of vertex N-magic total labeling graphs. In terms of magic constants k i for i ∈ {1, 2, ..., N }, we also found the requirement for vertex N-magic total labeling of trees. We investigated the potential for vertex N-magic total labeling at vertices in graphs with varying vertex degrees.
$V_k$-Super vertex magic labeling of 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.