A survey of face-antimagic evaluations of graphs

2016

Let ( , , ) G V E F  be a simple, finite, plane graph with |V (G)| number of vertices, |E (G)| number of edges and |F (G)| number of faces. A labeling of type (1,1,0) is a bijective mapping from the vertex set and edge set of G to the set {1, 2,..., |V (G)| + |E (G)|}. Moreover, the labeling is called super if the vertices are labeled with the smallest numbers. The weight of a face under the labeling is the sum of labels of edges and vertices surrounding that face. In this paper we study face bimagic labeling of type (1,1,0) for wheels, cylinders and disjoint union of m copies of prism graphs.

An (a, d)-edge-antimagic total labeling of G is a one-to-one mapping f taking the vertices and edges onto 1, 2, . . . ,

Applied Mathematics & Information Sciences, 2015

A face irregular entire k-labeling ϕ : V ∪ E ∪ F → {1, 2,. .. , k} of a 2-connected plane graph G = (V, E, F) is a labeling of vertices, edges and faces of G in such a way that for any two different faces f and g their weights w ϕ (f) and w ϕ (g) are distinct. The weight of a face f under a k-labeling ϕ is the sum of labels carried by that face and all the edges and vertices incident with the face. The minimum k for which a plane graph G has a face irregular entire k-labeling is called the entire face irregularity strength. We investigate a face irregular entire labeling as a modification of the well-known vertex irregular and edge irregular total labelings of graphs. We obtain some estimations on the entire face irregularity strength and determine the precise values for graphs from three families of plane graphs.

Australas. J Comb., 2007

Discussiones Mathematicae Graph Theory, 2020

We investigate face irregular labelings of plane graphs and we introduce new graph characteristics, namely face irregularity strength of type (α, β, γ). We obtain some estimation on these parameters and determine the precise values for certain families of plane graphs that prove the sharpness of the lower bounds.

2005

A bijection mapping that assigns natural numbers to vertices and/or edges of a graph is called a labeling. In this thesis, we consider graph labelings that have weights associated with each edge and/or vertex. If all the vertex weights (respectively, edge weights) have the same value then the labeling is called magic. If the weight is different for every vertex (respectively, every edge) then we called the labeling antimagic. Since its introduction by Sedláček in 1963, research in both magic and antimagic labelings has been growing fast. In this thesis we introduce some variations of magic and antimagic labelings and discuss their properties and provide corresponding labeling schemes. There are two main parts in this thesis. One main part is on vertex labeling and the other main part is on edge labeling. In vertex labeling, we introduce super vertex magic total labeling and super (a, d)vertex antimagic total labeling, and present new results in these labelings. Additionally, we give vertex labeling schemes for some particular classes of graphs. Moreover, we introduce consecutive vertex magic total labeling and give new results concerning properties of consecutive vertex magic total labeling. In edge labeling, we present new results in super (a, d)-edge antimagic total labeling. We go on to introduce a consecutive edge magic total labeling and present some new vii viii results in this labeling. As in the vertex labeling chapter, we also give edge labeling schemes for some particular classes of graphs.

Mathematics and Statistics, 2025

A directed graph D can be labeled antimagically by assigning different integers to its arcs, ensuring that the computed vertex weights are distinct. Antimagic labeling exists in a digraph D with h arcs and g vertices if it is possible to uniquely match each arc to an integer from 1 to h. For every vertex, the difference between the totals of the labels from incoming arcs and outgoing arcs is unique for each vertex. A directed graph that allows such antimagic labeling is referred to as an antimagic digraph. There are countless methods to create an antimagic digraph. These constructions are essential in fields such as network theory, coding theory, and combinatorial optimization. The subset sum problem is a well-recognized issue in both computer science and combinatorics. These problems play a decisive role in graph labeling, especially when it comes to the creation and evaluation of specific types of labels. In this paper, we connect the idea of subset sum problems with wheel digraphs-→ W n to represent it as antimagic. The Cartesian product of directed graphs is a fundamental concept in graph theory that holds both theoretical and practical importance. The applications of Cartesian products include network design and analysis, parallel computing, graph decomposition and construction, Game Theory, and Decision-Making. Additionally, in this paper, we have developed antimagic digraphs from the Cartesian product of directed path P n and-→ K 2 by traversing the directed path-→ P n in alternating and unidirectional ways.

Discussiones Mathematicae Graph Theory, 2003

In this paper we introduce a new type of graph labeling for a graph G(V, E) called an (a, d)-vertex-antimagic total labeling. In this labeling we assign to the vertices and edges the consecutive integers from 1 to |V | + |E| and calculate the sum of labels at each vertex, i.e., the vertex label added to the labels on its incident edges. These sums form an arithmetical progression with initial term a and common difference d.

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 vertex u is the sum of labels of all edges incident to such vertex. A graph is called anti-magic if it admits an antimagic labeling. Hartsfield and Ringel conjectured in 1990 that all connected graphs except K 2 are anti-magic. Recently, N. Alon et al (Dense graphs are anti-magic,

An antimagic labeling of a graph G is a one-to-one correspondence between E(G) and {1, 2, . . . , |E|} such that for any two distinct vertices u and v, the sum of the labels assigned to edges incident to u is distinct from the sum of the labels assigned to edges incident to v. It was conjectured by Hartsfield and Ringel [9] in 1990 that every connected graph other than K 2 has an antimagic labeling. This conjecture attracted considerable attention.