Andrea Semaničová-Feňovčíková

ON CLASSES OF REGULAR GRAPHS WITH CONSTANT METRIC DIMENSION

Abstract In this paper, we are dealing with the study of the metric dimension of some classes of regular graphs by considering a class of bridgeless cubic graphs called the flower snarks J n , a class of cubic convex polytopes considering the open problem raised in [M. Imran et al., families of plane graphs with constant metric dimension, Utilitas Math., in press] and finally Harary graphs H 5, n by partially answering to an open problem proposed in [I. Javaid et al., Families of regular graphs with constant metric dimension, Utilitas Math., 2012, 88: 43–57]. We prove that these classes of regular graphs have constant metric dimension. It is natural to ask for the characterization of regular graphs with constant metric dimension.

On super (a,1)-edge-antimagic total labelings of regular graphs

A labeling of a graph is a mapping that carries some set of graph elements into numbers (usually positive integers). An (a,d)-edge-antimagic total labeling of a graph with p vertices and q edges is a one-to-one mapping that takes the vertices and edges onto the integers 1,2...,p+q, so that the sum of the labels on the edges and the labels of their end vertices forms an arithmetic progression starting at a and having difference d. Such a labeling is called super if the p smallest possible labels appear at the vertices. In this paper we prove that every even regular graph and every odd regular graph with a 1-factor are super (a,1)-edge-antimagic total. We also introduce some constructions of non-regular super (a,1)-edge-antimagic total graphs.

Total Irregularity Strength of Three Families of Graphs

We deal with the totally irregular total labeling which is required to be at the same time vertex irregular total and also edge irregular total. The minimum k for which a graph G has a totally irregular total k-labeling is called the total irregularity strength of G. In this paper, we estimate the upper bound of the total irregularity strength of graphs and determine the exact value of the total irregularity strength for three families of graphs.

Wheels are cycle-antimagic

Abstract A simple graph G admits an H-covering if every edge in E ( G ) belongs to a subgraph of G isomorphic to H. An (a, d)-H-antimagic total labeling of a graph G admitting an H-covering is a bijective function from the vertex set V ( G ) and the edge set E ( G ) of the graph G onto the set of integers { 1 , 2 , … , | V ( G ) | + | E ( G ) | } such that for all subgraphs H ′ isomorphic to H, the sum of labels of all the edges and vertices belonging to H ′ constitute the arithmetic progression with the initial term a and the common difference d. Such a labeling is called super if the smallest possible labels appear on the vertices. In this paper, we investigate the existence of super cycle-antimagic total labelings of wheel.

Super-vertex-antimagic total labelings of disconnected graphs

Let G=(V,E) be a finite, simple and non-empty (p,q)-graph of order p and size q. An (a,d)-vertex-antimagic total labeling is a bijection f from V(G)@?E(G) onto the set of consecutive integers 1,2,...,p+q, such that the vertex-weights form an arithmetic progression with the initial term a and the common difference d, where the vertex-weight of x is the sum of values f(xy) assigned to all edges xy incident to vertex x together with the value assigned to x itself, i.e. f(x). Such a labeling is called super if the smallest possible labels appear on the vertices. In this paper, we will study the properties of such labelings and examine their existence for disconnected graphs.

On

An -vertex-antimagic edge labeling (or an -VAE labeling, for short) of is a bijective mapping from the edge set of a graph to the set of integers with the property that the vertex-weights form an arithmetic sequence starting from and having common difference , where and are two positive integers, and the vertex-weight is the sum of the labels of all edges incident to the vertex. A graph is called -antimagic if it admits an -VAE labeling. In this paper, we investigate the existence of -VAE labeling for disconnected 3-regular graphs. Also, we define and study a new concept -vertex-antimagic edge deficiency, as an extension of -VAE labeling, for measuring how close a graph is away from being an -antimagic graph. Furthermore, the -VAE deficiency of Hamiltonian regular graphs of even degree is completely determined. More open problems are mentioned in the concluding remarks.

(a,1)

Abstract This survey aims to give an overview of the modifications of the well-known irregular assignments, namely edge irregular labelings, vertex irregular and edge irregular total labelings, and face irregular entire labelings of graphs.

-Vertex-Antimagic Edge Labeling of Regular Graphs

A simple graph G admits an H-covering if every edge in E(G) belongs to a subgraph of G isomorphic to H. An (a, d)-H-antimagic total labeling of a graph G admitting an H-covering is a bijective function from the vertex set V(G) and the edge set E(G) of the graph G onto the set of integers {1, 2, . . . , |V (G)| + |E(G)|} such that for all subgraphs H’ isomorphic to H, the sum of labels of all the edges and vertices belonging to H’ constitute the arithmetic progression with the initial term a and the common difference d. Such a labeling is called super if the smallest possible labels appear on the vertices. In this paper, we study super (a,d)-H-antimagic labellings of a disjoint union of graphs for d=|E(H)|−|V(H)|

A Survey of Irregularity Strength

For a graph , a bijection from is called -edge-antimagic total (-EAT) labeling of if the edge-weights , form an arithmetic progression starting from and having a common difference , where and are two fixed integers. An -EAT labeling is called super -EAT labeling if the vertices are labeled with the smallest possible numbers; that is, . In this paper, we study super -EAT labeling of cycles with some pendant edges attached to different vertices of the cycle.

On h-antimagicness of disconnected graphs

Abstract A simple graph G = (V, E) admits an H-covering, if every edge in E(G) belongs to a subgraph of G isomorphic to H. A graph G admitting an H-covering is called an (a, d)-H-antimagic if there exists a bijective function f : V(G) ∪ E(G) → {1, 2, …, |V(G)| + |E(G)|} such that for all subgraphs H′ isomorphic to H the sums ∑v∈V(H′)f(v) + ∑e∈E(H′)f(e) form an arithmetic sequence {a, a + d, …, a + (t − 1)d}, where a > 0 and d ≥ 0 are integers and t is the number of all subgraphs of G isomorphic to H. Moreover, if the vertices are labeled with numbers 1, 2, …, |V(G)| the graph is called super. In this paper we deal with super cycle-antimagicness of subdivided graphs. We also prove that the subdivided wheel admits an (a, d)-cycle-antimagic labeling for some d.