Martin Bača

Technical University of Kosice - Technicka univerzita v Kosiciach · Department Applied Mathematics and Informatics

A (modular) vertex irregular total labeling of a graph $ G $ of order $ n $ is an assignment of positive integers from $ 1 $ to $ k $ to the vertices and edges of $ G $ with the property that all vertex weights are distinct. The vertex weight of a vertex $ v $ is defined as the sum of numbers assigned to the vertex $ v $ itself and to the edge's in...

A k-labeling from the vertex set of a simple graph G=(V,E) to a set of integers {1,2,⋯,k} is defined to be a modular edge irregular if, for every couple of distinct edges, their modular edge weights are distinct. The modular edge weight is the remainder of the division of the sum of end vertex labels by modulo |E(G)|. The modular edge irregularity...

A graceful labeling of a graph G is an injective function from the vertex set of G to the set \(\{0,1,\dots ,|E(G)|\}\) such that the induced edge labels are all different, where an induced edge label is defined as the absolute value of the difference between the labels of its end vertices. If the induced edge labeling is simultaneously antimagic,...

A fullerene is a polyhedral molecule made entirely of carbon atoms other than graphite and diamond. Mathematical chemistry or Chemical graph theory as a combination of chemistry and graph theory studies the physical properties of mathematically modeled chemical compounds. Graph labeling is a vital area of graph theory that has applications in both...

Let G = (V, E) be a finite simple undirected graph without K2 components. A bijection f : E → {1, 2, ⋯, |E|} is called a local antimagic labeling if for any two adjacent vertices u and v, they have different vertex sums, i.e., w(u) ≠ w(v), where the vertex sum w(u) = ∑e∈E(u) f(e), and E(u) is the set of edges incident to u. Thus any local antimagic...

Let $G = (V, E)$ be a finite simple undirected graph without $K_2$ components. A bijection $f : E \rightarrow \{1, 2,\cdots, |E|\}$ is called a local antimagic labeling if for any two adjacent vertices $u$ and $v$, they have different vertex sums, i.e., $w(u) \neq w(v)$, where the vertex sum $w(u) = \sum_{e \in E(u)} f(e)$, and $E(u)$ is the set of...

If every edge in the graph G is also an edge of a subgraph of G isomorphic to a given graph H we say that the graph G admits an H-covering. Let G be a graph admitting an H-covering and let H1, H2, … , Ht be all subgraphs in G isomorphic to H. The edge H-irregularity strength of a graph G is the smallest integer k for which one can find a mapping ϕ...

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.

For a simple graph $ G = (V, E) $ with the vertex set $ V(G) $ and the edge set $ E(G) $, a vertex labeling $ \varphi: V(G) \to \{1, 2, \dots, k\} $ is called a $ k $-labeling. The weight of an edge under the vertex labeling $ \varphi $ is the sum of the labels of its end vertices and the modular edge-weight is the remainder of the division of this...

Metric dimension and fault-tolerant metric dimension of any graph G is subject to size of resolving set. It has become more important in modern GPS and sensors based world as resolving set ensures that in case of semi outage system is still scalable using redundant interfaces. Metric dimension of several interesting classes of graphs have been inve...

It is easily observed that the vertices of a simple graph cannot have pairwise distinct degrees. This means that no simple graph of the order of at least two is, in this way, irregular. However, a multigraph can be irregular. Chartrand et al., in 1988, posed the following problem: in a loopless multigraph, how can one determine the fewest parallel...

Let G=(V,E) be a simple graph. A vertex labeling f:V(G)→{1,2,⋯,k} is defined to be a local inclusive (respectively, non-inclusive) d-distance vertex irregular labeling of a graph G if for any two adjacent vertices x,y∈V(G) their weights are distinct, where the weight of a vertex x∈V(G) is the sum of all labels of vertices whose distance from x is a...

An edge labeling of a graph G=(V,E) using every label from the set {1,2,⋯,|E(G)|} exactly once is a local antimagic labeling if the vertex-weights are distinct for every pair of neighboring vertices, where a vertex-weight is the sum of labels of all edges incident with that vertex. Any local antimagic labeling induces a proper vertex coloring of G...

For a simple graph G with no isolated edges and at most, one isolated vertex, a labeling φ:E(G)→{1,2,…,k} of positive integers to the edges of G is called irregular if the weights of the vertices, defined as wtφ(v)=∑u∈N(v)φ(uv), are all different. The irregularity strength of a graph G is known as the maximal integer k, minimized over all irregular...

Graph labeling is the mapping of elements of a graph (which can be vertices, edges, faces or a combination) to a set of numbers. The mapping usually produces partial sums (weights) of the labeled elements of the graph, and they often have an asymmetrical distribution. In this paper, we study vertex–face and edge–face labelings of two-connected plan...

We introduce a modular irregularity strength of graphs as modification of the well-known irregularity strength. We obtain some estimation on modular irregularity strength and determine the exact values of this parameter for five families of graphs.

The applications of finite commutative ring are useful substances in robotics and programmed geometric, communication theory, and cryptography. In this paper, we study the vertex-based eccentric topological indices of a zero-divisor graphs of commutative ring , where and are primes. 1. Introduction One of the most significant issues in science is...

A digraph D is called (a, d)-vertex-in-antimagic ((a, d)-vertex-out-antimagic) if it is possible to label its vertices and arcs with distinct numbers from the set {1, 2, … ,|V(D)|+|A(D)|} such that the set of all in-vertex-weights (out-vertex-weights) form an arithmetic sequence with initial term a and a common difference d, where a > 0, d ≥ 0 are...

Let be a finite simple graph with p vertices and q edges. A decomposition of a graph G into isomorphic copies of a graph H is called (a, d)-H-antimagic if there is a bijection such that for all subgraphs isomorphic to H in the decomposition of G, the sum of the labels of all the edges and vertices belonging to constitutes an arithmetic progression...

A simple graph G = (V(G),E(G)) admits an H-covering if every edge in E(G) belongs to at least one subgraph of G isomorphic to a given graph H. A total k-labeling φ : V(G) ∪ E(G) → {1,2,..., k} is called to be an H-irregular total k-labeling of the graph G admitting an H-covering if for every two different subgraphs H' and H" isomorphic to H there i...

A set of vertices W is a resolving set of a graph G if every two vertices of G have distinct representations of distances with respect to the set W. The number of vertices in a smallest resolving set is called the metric dimension. This invariant has extensive applications in robotics, since the metric dimension can represent the mínimum number of...

Let D be a digraph, possibly with loops. A queen labeling of D is a bijective function \(l:V(G)\longrightarrow \{1,2,\ldots ,|V(G)|\}\) such that, for every pair of arcs in E(D), namely (u, v) and \((u',v')\) we have (i) \(l(u)+l(v)\ne l(u')+l(v')\) and (ii) \(l(v)-l(u)\ne l(v')-l(u')\). Similarly, if the two conditions are satisfied modulo \(n=|V(...

A simple graph G = (V,E) admits an H-covering if every edge in E belongs to at least one subgraph of G isomorphic to a given graph H. The graph G admitting an H-covering is (a, d)-H-antimagic if there exists a bijection f : V ∪E → {1, 2, . . . , |V |+|E|} such that, for all subgraphs H′ of G isomorphic to H, the H′-weights, wtf (H′) = Σ v∈V (H′) f(...

An edge-colored graph G is rainbow k-connected, if for every two vertices of G, there are k internally disjoint rainbow paths, i.e., if no two edges of each path are colored the same. The minimum number of colors needed for which there exists a rainbow k-connected coloring of G, \(rc_k(G)\), is the rainbow k-connection number of G. Let G and H be t...

Borchert and Gosselin et al. solved the problem of finding metric dimension for Harary graph H4,n; n at least 8. In this paper, we find the minimal doubly resolving set, and hence the cardinality of this set for Harary graph.

A simple graph G admits an H-covering if every edge in E(G) belongs to at least to one subgraph of G isomorphic to a given graph H. For the subgraph H ⊆ G under a total k-labeling we define the associated H-weight as the sum of labels of all vertices and edges belonging to H. The total k-labeling is called the H-irregular total k-labeling of a grap...

In this paper we prove that complete graphs admit totally antimagic total labeling. We also deal with the problem of finding a total labeling for prism and for two special classes of graphs related to path that are simultaneously edge-magic and vertex-antimagic.

The final chapter opens with a brief summary of the book. This is followed by a collection of conjectures and problems that, at the time of writing, were still unsolved. To the interested researcher this would undoubtedly be the most valuable chapter in the book.

This chapter is devoted to the study of vertex magic total labelings. Constructions are given for regular and non-regular graphs as well as for some standard graph families. These labelings are also explored for disjoint families of particular graphs.

This chapter explores the relationship between antimagic labeling and alpha labelings and also the well-known graceful labelings. Much of this chapter looks at interesting labelings and structures on trees, including edge antimagic trees, alpha trees, and disjoint union of caterpillars.

After vertex magic total labelings, this chapter has a focus on edge magic total labelings. Labeling schemes are given for connected and disconnected graphs in addition to well-known graph families. Strong super edge magic labelings are introduced and relationships between super edge magic total labelings and other labeling schema are explored.

Following the chapters on magic type labelings, this chapter begins the section of the book devoted to antimagic labelings. Vertex antimagic and super vertex antimagic labelings, both edge labels and total labels are investigated with labeling constrictions given for connected and disconnected graphs.

This chapter introduces magic and supermagic graphs giving relevant definitions and tracing the evolution of magic graphs from its genesis in magic squares. The text proceeds to study magic and supermagic labelings on particular types of graphs and looks at the impact of various graph operations.

This chapter focuses on edge-antimagic graphs under both vertex labelings and total labelings. Super edge-antimagic total labelings are given for standard graphs and (a,1) edge-antimagic total labelings are introduced and explored.

We investigate modifications of the well-known irregularity strength of graphs, namely, total (vertex, edge) H-irregularity strengths. Recently the bounds and precise values for some families of graphs concerning these parameters have been determined. In this paper, we determine the exact value of the total (vertex, edge) H-irregularity strengths f...

For a graph G an edge-covering of G is a family of subgraphs H1, H2, . . . , Ht such that each edge of E(G) belongs to at least one of the subgraphs Hi, i = 1, 2, . . . , t. In this case we say that G admits an (H1, H2, . . . , Ht)-(edge) covering. An H-covering of graph G is an (H1, H2, . . . , Ht)-(edge) covering in which every subgraph Hi is iso...

Magic and antimagic labelings are among the oldest labeling schemes in graph theory. This book takes readers on a journey through these labelings, from early beginnings with magic squares up to the latest results and beyond. Starting from the very basics, the book offers a detailed account of all magic and antimagic type labelings of undirected gra...

For a simple graph G=(V,E) with the vertex set V and the edge set E and for an integer k, 2≤k≤|E(G)|, an edge labeling φ:E(G)→{0,1,…,k−1} induces a vertex labeling φ∗:V(G)→{0,1,…,k−1} defined by φ∗(v)=φ(e1)⋅φ(e2)⋅…⋅φ(en)(modk), where e1,e2,…,en are the edges incident to the vertex v. The function φ is called a k-total edge product cordial labeling...

For a graph we define -labeling such that the edges of are labeled with integers and the vertices of are labeled with even integers , where . The labeling is called a vertex irregular reflexive-labeling if distinct vertices have distinct weights, where the vertex weight is defined as the sum of the label of that vertex and the labels of all edges i...

For a simple graph G = (V,E) this paper deals with the existence of an edge labeling ψ: E(G) → {0, 1, . . ., k-1}, 2 ≤ k ≤ |E(G)|, which induces a vertex labeling ψ*: V (G) → {0; 1;: :: ; k-1} in such a way that for each vertex v, assigns the label ψ(e1) · ψ(e2) ·: :: · ψ(en) (mod k), where e1, e2, . . ., en are the edges incident to the vertex v....

For a simple graph G, a vertex labeling f: V (G) → (1; 2....,k) is called a k-labeling. The weight of a vertex v, denoted by wtf (v) is the sum of all vertex labels of vertices in the closed neighborhood of the vertex v. A vertex k-labeling is defined to be an inclusive distance vertex irregular distance k-labeling of G if for every two different v...

A vertex $z$ in a connected graph $G$ \textit{resolves} two vertices $u$ and $v$ in $G$ if $d_G(u,z)\neq d_G(v,z)$. \ A set of vertices $R_G\{u,v\}$ is a set of all resolving vertices of $u$ and $v$ in $G$. \ For every two distinct vertices $u$ and $v$ in $G$, a \textit{resolving function} $f$ of $G$ is a real function $f:V(G)\rightarrow[0,1]$ such...

For a graph G, an edge labeling fe:E(G)→(1,2,...,ke) and a vertex labeling fv:V(G)→(0,2,4,...,2kv) are called total k-labeling, where k=max(ke,2kv). The total k-labeling is called an edge irregular reflexive k -labeling of the graph G, if for every two different edges xy and x'y' of G, one has wt(xy)=fv(x)+fe(xy)+fv(y)≠wt(x'y')=fv(x')+fe(x'y')+fv(y...

A graph G = (V (G), E(G)) admits an H-covering if every edge in E belongs to a subgraph of G isomorphic to H. A graph G admitting an H-covering is called (a; d) -H-antimagic if there is a bijection f: V (G) (n-ary union) E(G) → (1; 2,..,|V(G)| + |E(G)|) such that, for all subgraphs H' of G isomorphic to H, the H-weights, wtf (H') =∑ υ∈V (H') f(v)+...

A simple graph G admits an H-covering if every edge in E(G) belongs to at least to one subgraph of G isomorphic to a given graph H. For the subgraph H ⊆ G under a total k-labeling we define the associated H-weight as the sum of labels of all vertices and edges belonging to H. The total k-labeling is called the H-irregular total k-labeling of a grap...

Consider a simple connected undirected graph G = (V;E), where V represents the vertex set and E represents the edge set, respectively. A subset D of V is called doubly resolving set if for every two vertices x; y of G, there are two vertices u; v ∈ D such that d(u; x) - d(u; y) ≠= d(v; x) - d(v; y). A doubly resolving set with minimum cardinality i...

Algorithms help in solving many problems, where other mathematical solutions are very complex or impossible. Computations help in tackling numerous issues, where other numerical arrangements are extremely perplexing or incomprehensible. In this paper, the edge irregularity strength of a complete binary tree (T2,h), complete ternary tree (T3,h) and...

For any two vertices u, v in a connected graph G, the interval I(u, v) consists of all vertices which are lying in some u − v shortest path in G. A vertex x in a graph G strongly resolves a pair of vertices u, v if either u ∈ I(x, v) or v ∈ I(x, u). A set of vertices W of V (G) is called a strong resolving set if every pair of vertices of G is stro...

We study an edge irregular reflexive k-labelling for the generalized friendship graphs, also known as flowers (a symmetric collection of cycles meeting at a common vertex), and determine the exact value of the reflexive edge strength for several subfamilies of the generalized friendship graphs.

For any given two graphs G and H, the notation \(F\rightarrow \) (G, H) means that for any red–blue coloring of all the edges of F will create either a red subgraph isomorphic to G or a blue subgraph isomorphic to H. A graph F is a Ramsey (G, H)-minimal graph if \(F\rightarrow \) (G, H) but \(F-e\nrightarrow (G,H)\), for every \(e \in E(F)\). The c...

In this paper, we continue investigating the partition dimension for disconnected graphs. We determine the partition dimension for some classes of disconnected graphs G consisting of two components. If \(G=G_1 \cup G_2\), then we give the bounds of the partition dimension of G for \(G_1 = P_n\) or \(G_1=C_n\) and also for \(pd(G_1)=pd(G_2)\).

A simple graph G = (V (G),E(G)) admits an H-covering if every edge in E(G) belongs at least to one subgraph of G isomorphic to a given graph H. Then the graph G admitting H- covering admits an H-irregular total k-labeling f: V (G)∪E(G) → (1, 2, . . ., k) if for every two different subgraphs H' and H'' isomorphic to H there is wtf (H') ≠ wtf (H'' P)...

The concept of face-antimagic labeling of plane graphs was introduced by Mirka Miller in 2003. This survey aims to give an overview of the recent results obtained in this topic.

In this paper we investigate the total edge irregularity strength tes(G) and the total vertex irregularity strength tvs(G) of diamond graphs Brn and prove that tes(Brn)=(5n-3)/3, while tvs(Brn)=(n+1)/3.

An edge irregular total $k$-labeling of a~graph $G=(V,E)$ is a~labeling $\varphi: V\cup E \to \{ 1,2, \dots, k \}$ such that the total edge-weights $wt(ab)= \varphi(a)+ \varphi(ab) + \varphi(b)$ are different for all pairs of distinct edges. The minimum $k$ for which the graph $G$ has an~edge irregular total $k$-labeling is called the {\it total ed...

A simple graph $G=(V,E)$ admits an~$H$-covering if every edge in $E$ belongs at least to one subgraph of $G$ isomorphic to a given graph $H$. An {\it $(a,d)$-$H$-antimagic labeling} of $G$ admitting an $H$-covering is a~bijective function $f: V\cup E \to \{1,2,\dots, |V|+|E| \}$ such that, for all subgraphs $H'$ of $G$ isomorphic to $H$, the $H'$-w...

New graph characteristic, the total H-irregularity strength of a graph, is introduced. Estimations on this parameter are obtained and for some families of graphs the precise values of this parameter are proved.

A face irregular entire k-labeling of a 2-connected plane graph G is a labeling of vertices, edges and faces of G with labels from the set {1,2,…,k} in such a way that for any two different faces their weights are distinct. The weight of a face under a k-labeling is the sum of labels carried by that face and all the edges and vertices incident with...

In this paper we give a survey on several types of colourings of elements of graphs by different types of labellings.

The {\it generalized prism} $P_n^m$ can be defined as the Cartesian product of a~cycle on $n$ vertices with a~path on $m$ vertices. In this paper we deal with the problem of labeling the vertices, edges and faces of a disjoint union of r copies of P_n^m by the consecutive integers starting from 1 in such a way that the set of face-weights of all s...

Let $G$ be a connected graph‎. ‎Let $f$ be a proper $k$-coloring of $G$ and $\Pi=\{R_1,R_2,\ldots‎, ‎R_k\}$ be an ordered partition of $V(G)$ into color classes‎. ‎For any vertex $v$ of $G,$ define the {\em color code} $c_\Pi(v)$ of $v$ with respect to $\Pi$ to be a $k$-tuple $(d(v,R_1),d(v,R_2),\ldots,d(v,R_k)),$ where $d(v,R_i)= \text{min}\{d(v,x...

Let H be a graph. A graph G=(V,E) admits an H-covering if every edge in E belongs to a subgraph of G isomorphic to H. A graph G admitting an H-covering is called (a,d)-H-antimagic if there is a bijection f:V(G)∪E(G)→{1,2,…,|V(G)|+|E(G)|} such that for each subgraph H′ of G isomorphic to H, the sum of labels of all the edges and vertices belonged to...

The Klein-bottle fullerene is a finite trivalent graph embedded on the Klein-bottle such that each face is a hexagon. The paper deals with the problem of labeling the vertices, edges and faces of the Klein-bottle fullerene in such a way that the label of a face and the labels of the vertices and edges surrounding that face add up to a weight of tha...

Let $D$ be a digraph, possibly with loops. A queen labeling of $D$ is a bijective function $l:V(G)\longrightarrow \{1,2,\ldots,|V(G)|\}$ such that, for every pair of arcs in $E(D)$, namely $(u,v)$ and $(u',v')$ we have (i) $l(u)+l(v)\neq l(u')+l(v')$ and (ii) $l(v)-l(u)\neq l(v')-l(u')$. Similarly, if the two conditions are satisfied modulo $n=|V(G...

Let \(G=(V,E)\) be a connected graph with \(\left| V \right| =n\) and \(\left| E \right| = m.\) A bijection \(f:E \rightarrow \{1,2, \dots , m\}\) is called a local antimagic labeling if for any two adjacent vertices u and v, \(w(u)\ne w(v),\) where \(w(u)=\sum \nolimits _{e\in E(u)}{f(e)},\) and E(u) is the set of edges incident to u. Thus any loc...

For a graph G=(V(G),E(G)), an edge labeling ϕ:E(G)→(0,1,...,k-1) where k is an integer, 2≤k≤|E(G)|, induces a vertex labeling ϕ*:V(G)→(0,1,...,k-1) defined by ϕ*(v)=ϕ(e1)ϕ(e2)...ϕ(en)(modk), where e1,e2,...,en are the edges incident to the vertex v. The function ϕ is called a k-total edge product cordial labeling of G if |(eϕ(i)+vϕ*(i))-(eϕ(j)+vϕ*(...

A simple graph G = (V,E) admits an H-covering if every edge in E belongs at least to one subgraph of G isomorphic to a given graph H. Then the graph G admitting an H-covering is (a, d)-H-antimagic if there exists a bijection f : V ∪ E → {1, 2, . . . , |V | + |E|} such that, for all subgraphs H� of G isomorphic to H, the H�-weights, wtf (H� � ) = v∈...

An edge irregular k-labeling of a graph G is a labeling of the vertices of G with labels from the set {1, 2,...k} in such a way that for any two different edges xy and x'y' their weights w(xy) and w(x'y') are distinct. The weight w(xy) of an edge xy in G is the sum of the labels of the end vertices x and y. The minimum k for which the graph G has a...

A simple graph G = (V; E) admits an H-covering if every edge in E belongs to at least one subgraph of G isomorphic to a given graph H. Then the graph G is (a; d)-H-antimagic if there exists a bijection f : ∪ E → {1, 2, V + E} such that, for all subgraphs H0 of G isomorphic to H, the H-weights, wtf (H) =∑vv(H') f(v)+∑e∈(H') f(e) form an arithmetic p...

We introduce two new graph characteristics, the edge $H$-irregularity strength and the vertex $H$-irregularity strength. We estimate the bounds of these parameters and determine their exact values for several families of graphs namely, paths, ladders and fans.

For a simple graph G = (V,E) with the vertex set V and the edge set E, a vertex irregular total k-labeling f: V ∪E → {1, 2,…, k} is a labeling of vertices and edges of G in such a way that for any two different vertices x and x′, their weights wtf (x) = f(x) + ∑xy∈E f(xy) and wtf (x′) = f(x′)+ ∑x′y′∈E f(x′y′) are distinct. A smallest positive integ...

We deal with the modifications of the well-known irregular assignments, namely vertex irregular total labelings, edge irregular total labelings and totally irregular total labelings of graphs. In the paper, we study the total vertex (edge) irregularity strength and total irregularity strength for disjoint union of arbitrary graphs and we establish...

A molecular graph is a graph in which vertices are atoms of a given molecule and edges are its chemical bonds. A numerical quantity which characterizes the whole structure of a graph is called a topological index. Three degree based topological indices, the Randić (Rα ), the atombond connectivity (ABC) and the geometric-arithmetic (GA) indices of m...

A numerical quantity that characterizes the whole structure of a graph is called a topological index. The concept of Randic (R-alpha), atom-bond connectivity (ABC), and geometric-arithmetic (GA) topological indices was established in chemical graph theory based on vertex degrees. In this paper, we study a carbon nanotube network that is motivated b...

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...