Distance magic labelings of product graphs (original) (raw)

Distance magic labelings of graphs

2003

As a natural extension of previously defined graph labelings, we introduce in this paper a new magic labeling whose evaluation is based on the neighbourhood of a vertex. We define a 1-vertex-magic vertex labeling of a graph with v vertices as a bijection f taking the vertices to the integers 1, 2,. .. , v with the property that there is a constant k such that at any vertex x, y∈N (x) f (y) = k, where N (x) is the set of vertices adjacent to x. We completely solve the existence problem of 1-vertex-magic vertex labelings for all complete bipartite, tripartite and regular multipartite graphs, and obtain some non-existence results for other natural families of graphs.

Distance magic labelings of a union of graphs

A 1-vertex-magic vertex labeling of a graph G(V,E) with p vertices is a bijection f from the vertex set V(G) to the integers 1,2,⋯,p with the property that there is a constant k such that at any vertex x the sum ∑f(x) taken over all neighbors of x is k. In this paper, we study the 1-vertex-magic vertex labelings of two families of disconnected graphs, namely a disjoint union of m copies of complete p-partite graph and a disjoint union of m copies of 2n-regular graph C p [K n ¯].

Magic labelings of distance at most 2

For an arbitrary set of distances D ⊆ {0, 1, . . . , d}, a graph G is said to be Ddistance magic if there exists a bijection f : V → {1, 2, . . . , v} and a constant k such that for any vertex

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.

Union of distance magic graphs

Discussiones Mathematicae Graph Theory

A distance magic labeling of a graph G = (V, E) with |V | = n is a bijection ℓ from V to the set {1,. .. , n} such that the weight w(x) = y∈NG(x) ℓ(y) of every vertex x ∈ V is equal to the same element µ, called the magic constant. In this paper, we study unions of distance magic graphs as well as some properties of such graphs.

On anti-magic labeling for graph products

Discrete Mathematics, 2008

An anti-magic labeling of a finite simple undirected graph with p vertices and 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 one vertex is the sum of labels of all edges incident to such vertex. A graph is called anti-magic if it admits an anti-magic labeling. Hartsfield and Ringel conjectured in 1990 that all connected graphs except K 2 are anti-magic. Recently, Alon et al. showed that this conjecture is true for dense graphs, i.e. it is true for p-vertex graphs with minimum degree (log p). In this article, new classes of sparse anti-magic graphs are constructed through Cartesian products and lexicographic products.

1932 Union of Distance Magic Graphs

2016

A distance magic labeling of a graph G = (V,E) with |V | = n is a bijection l from V to the set {1, . . . , n} such that the weight w(x) = ∑ y∈NG(x) l(y) of every vertex x ∈ V is equal to the same element μ, called the magic constant. In this paper, we study unions of distance magic graphs as well as some properties of such graphs.

A Survey of Distance Magic Graphs

2014

In this report, we survey results on distance magic graphs and some closely related graphs. A distance magic labeling of a graph G with magic constant k is a bijection from the vertex set to {1, 2, . . . , n}, such that for every vertex x ∑ y∈NG(x) (y) = k, where NG(x) is the set of vertices of G adjacent to x. If the graph G has a distance magic labeling we say that G is a distance magic graph. In Chapter 1, we explore the background of distance magic graphs by introducing examples of magic squares, magic graphs, and distance magic graphs. In Chapter 2, we begin by examining some basic results on distance magic graphs. We next look at results on different graph structures including regular graphs, multipartite graphs, graph products, join graphs, and splitting graphs. We conclude with other perspectives on distance magic graphs including embedding theorems, the matrix representation of distance magic graphs, lifted magic rectangles, and distance magic constants. In Chapter 3, we st...

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 distance labelings of 2-regular graphs

Electronic Journal of Graph Theory and Applications, 2021

Let G be a graph with |V (G)| vertices and ψ : V (G) −→ {1, 2, 3, • • • , |V (G)|} be a bijective function. The weight of a vertex v ∈ V (G) under ψ is w ψ (v) = u∈N (v) ψ(u). The function ψ is called a distance magic labeling of G, if w ψ (v) is a constant for every v ∈ V (G). The function ψ is called an (a, d)-distance antimagic labeling of G, if the set of vertex weights is a, a + d, a + 2d,. .. , a + (|V (G)| − 1)d. A graph that admits a distance magic (resp. an (a, d)distance antimagic) labeling is called distance magic (resp. (a, d)-distance antimagic). In this paper, we characterize distance magic 2-regular graphs and (a, d)-distance antimagic some classes of 2-regular graphs.