Induced Vertex Labeling Research Articles

Complete characterization of bridge graphs with local antimagic chromatic number 2

An edge labeling of a connected graph $G = (V, E)$ is said to be local antimagic if it is a bijection $f\colon E \to\{1,\ldots ,|E|\}$ such that for any pair of adjacent vertices $x$ and $y$, $f^+(x)\not= f^+(y)$, where the induced vertex label $f^+(x)= \sum f(e)$, with $e$ ranging over all the edges incident to $x$. The local antimagic chromatic number of $G$, denoted by $\chi_{la}(G)$, is the minimum number of distinct induced vertex labels over all local antimagic labelings of $G$. In this paper, we characterize $s$-bridge graphs with local antimagic chromatic number 2.

On local antimagic chromatic numbers of circulant graphs join with null graphs or cycles

An edge labeling of a graph G = (V,E) is said to be local antimagic if there is a bijection f : E → {1,..., |E|} such that for any pair of adjacent vertices x and y, f +(x) ≠ f +(y), where the induced vertex label is f +(x) = 𝜮 f(e), with e ranging over all the edges incident to x. The local antimagic chromatic number of G, denoted by χla(G), is the minimum number of distinct induced vertex labels over all local antimagic labelings of G. For a bipartite circulant graph G, it is known that χ(G)=2 but χla(G) ≥ 3. Moreover, χla(Cn ∨ K1)=3 (respectively 4) if n is even (respectively odd). Let G be a graph of order m ≥ 3. In [Affirmative solutions on local antimagic chromatic number, Graphs Combin., 36 (2020), 1337—1354], the authors proved that if m ≡ n (mod 2) with χla(G) = χ(G), m>n ≥ 4 and m ≥ n2/2, then χla(G∨On) = χla(G)+1. In this paper, we show that the conditions can be omitted in obtaining χla(G∨H) for some circulant graph G, and H is a null graph or a cycle. The local antimagic chromatic number of certain wheel related graphs are also obtained.

On local antimagic chromatic number of cycle-related join graphs II

An edge labeling of a graph [Formula: see text] is said to be local antimagic if it is a bijection [Formula: see text] such that for any pair of adjacent vertices [Formula: see text] and [Formula: see text], [Formula: see text], where the induced vertex label of [Formula: see text] is [Formula: see text] ([Formula: see text] is the set of edges incident to [Formula: see text]). The local antimagic chromatic number of [Formula: see text], denoted by [Formula: see text], is the minimum number of distinct induced vertex labels over all local antimagic labelings of [Formula: see text]. In this paper, several sufficient conditions to determine the local antimagic chromatic number of the join of graphs are obtained. We then determine the exact value of the local antimagic chromatic number of many join graphs.

On the Integer-antimagic Spectra of Non-Hamiltonian Graphs

Let $A$ be a nontrivial abelian group. A connected simple graph $G = (V, E)$ is $A$-\textbf{antimagic} if there exists an edge labeling $f: E(G) \to A \setminus \{0\}$ such that the induced vertex labeling $f^+: V(G) \to A$, defined by $f^+(v) = \Sigma$ $\{f(u,v): (u, v) \in E(G) \}$, is a one-to-one map. In this paper, we analyze the group-antimagic property for Cartesian products, hexagonal nets and theta graphs.

Constructing integer-magic graphs via the Combinatorial Nullstellensatz

Let A be a nontrivial abelian group and A* = A \ {0}. A graph is A-magic if there exists an edge labeling f using elements of A* which induces a constant vertex labeling of the graph. Such a labeling f is called an A-magic labeling and the constant value of the induced vertex labeling is called an A-magic value. In this paper, we use the Combinatorial Nullstellensatz to construct nontrivial classes of ℤp-magic graphs, prime p ≥ 3. For these graphs, some lower bounds on the number of distinct ℤp-magic labelings are also established.

Approaches that output infinitely many graphs with small local antimagic chromatic number

An edge labeling of a connected graph [Formula: see text] is said to be local antimagic if it is a bijection [Formula: see text] such that for any pair of adjacent vertices x and y, [Formula: see text], where the induced vertex label [Formula: see text], with e ranging over all the edges incident to x. The local antimagic chromatic number of G, denoted by [Formula: see text], is the minimum number of distinct induced vertex labels over all local antimagic labelings of G. In this paper, we show the existence of infinitely many bipartite and tripartite graphs with [Formula: see text].

On local antimagic chromatic number of spider graphs

An edge labeling of a connected graph G = (V, E) is said to be local antimagic if it is a Bijection f : E → {1, … ,|E|} such that for any pair of adjacent vertices x and y, f +(x) ≠ f +(y), where the induced vertex label f +(x) = Σf(e), with e ranging over all the edges incident to x. The local antimagic chromatic number of G, denoted by χla (G), is the minimum number of distinct induced vertex labels over all local antimagic labelings of G. In this paper, we first show that a d-leg spider graph has d + 1 ≤ χla ≤ + 2. We then obtain many sufficient conditions such that both the values are attainable. Finally, we show that each 3-leg spider has χla = 4 if not all legs are of odd length. No 3-leg spider with all odd leg lengths and χla = 5 is found. This provides partial solutions to the characterization of k-pendant trees T with χla (T) = k + 1 or k + 2. We conjecture that almost all d-leg spiders of size q that satisfy d(d + 1) ≤ 2(2q - 1) with each leg length at least 2 has χla = d + 1.

Application of the Combinatorial Nullstellensatz to Integer-magic Graph Labelings

Let $A$ be a nontrivial abelian group and $A^* = A \setminus \{0\}$. A graph is $A$-magic if there exists an edge labeling $f$ using elements of $A^*$ which induces a constant vertex labeling of the graph. Such a labeling $f$ is called an $A$-magic labeling and the constant value of the induced vertex labeling is called an $A$-magic value. In this paper, we use the Combinatorial Nullstellensatz to show the existence of $\mathbb{Z}_p$-magic labelings (prime $p \geq 3$ ) for various graphs, without having to construct the $\mathbb{Z}_p$-magic labelings. Through many examples, we illustrate the usefulness (and limitations) in applying the Combinatorial Nullstellensatz to the integer-magic labeling problem. Finally, we focus on $\mathbb{Z}_3$-magic labelings and give some results for various classes of graphs.

The integer-antimagic spectra of a disjoint union of Hamiltonian graphs

Let $A$ be a nontrivial abelian group. A simple graph $G = (V, E)$ is $A$-antimagic, if there exists an edge labeling $f: E(G) \to A \backslash \{0\}$ such that the induced vertex labeling $f^+(v)=\sum_{uv\in E(G)} f(uv)$ is a one-to-one map. The {integer-antimagic spectrum} of a graph $G$ is the set IAM$(G) = \{k: G {is} \mathbb{Z}_k{-antimagic and } k \geq 2\}$. In this paper, we determine the integer-antimagic spectra for a disjoint union of Hamiltonian graphs.

The integer-antimagic spectra of Hamiltonian graphs

Let A be a nontrivial abelian group. A connected simple graph G = ( V , E ) is A - antimagic , if there exists an edge labeling f : E ( G )→ A ∖ {0 A } such that the induced vertex labeling f + ( v )=∑ { u , v }∈ E ( G ) f ({ u , v }) is a one-to-one map. The integer-antimagic spectrum of a graph G is the set IAM ( G )={ k : G is ℤ k -antimagic and k ≥ 2} . In this paper, we determine the integer-antimagic spectra for all Hamiltonian graphs.

On number of pendants in local antimagic chromatic number

An edge labeling of a connected graph G = (V, E) is said to be local antimagic if it is a bijection f : E → {1, … ,|E|} such that for any pair of adjacent vertices x and y, f +(x) ≠ f +(y), where the induced vertex label f +(x) = ∑f(e), with e ranging over all the edges incident to x. The local antimagic chromatic number of G, denoted by χla (G), is the minimum number of distinct induced vertex labels over all local antimagic labelings of G. Let χ(G) be the chromatic number of G. In this paper, sharp upper and lower bounds of χla (G) for G with pendant vertices, and sufficient conditions for the bounds to equal, are obtained. Consequently, for k ≥ 1, there are infinitely many graphs with k ≥ χ(G) - 1 pendant vertices and χla (G) = k + 1. The work of this paper leads to conjecture and problem on graphs with pendant(s).

On modulo local antimagic chromatic number of graphs

An edge labeling of a connected (p, q) -graph G = (V, E) of order p and size q is a modulo local antimagic labeling if it is a bijection π : E → {1, … ,q} such that for any pair of adjacent vertices u and v, π +(u) ≠ π +(v), where the induced vertex label π +(u) = ∑π(e)(mod p), with e ranging over all the edges incident to u. The modulo local antimagic chromatic number of G, denoted πla (G), is the minimum number of distinct induced vertex labels over all modulo local antimagic labelings of G. In this paper, we shall study the relationship among chromatic number, local antimagic chromatic number and modulo local antimagic chromatic number of graphs. Many open problems for further research are also presented.

The Integer-antimagic Spectra of Graphs with a Chord

Let $A$ be a nontrival abelian group. A connected simple graph $G = (V, E)$ is $A$-antimagic if there exists an edge labeling $f: E(G) \to A \setminus \{0\}$ such that the induced vertex labeling $f^+: V(G) \to A$, defined by $f^+(v) = \sum_{uv\in E(G)}f(uv)$, is injective. The integer-antimagic spectrum of a graph $G$ is the set IAM$(G) = \{k\;|\; G \textnormal{ is } \mathbb{Z}_k\textnormal{-antimagic}$ $\textnormal{and } k \geq 2\}$. In this paper, we determine the integer-antimagic spectra for cycles with a chord, paths with a chord, and wheels with a chord.

EDGE ODD GRACEFUL LABELING OF SOME FLOWER PETAL GRAPHS

A labeling of a graph $G$ with $\alpha$ vertices and $\beta$ edges called an edge odd graceful labeling if there is an edge labeling with odd numbers to all edges such that each vertex is assigned a label which is the sum $\mod (2\gamma)$ of labels of edge incident on it, where $\gamma=max\{\alpha,\beta\}$ and the induced vertex labels are distinct. In this paper, we discussed about edge odd gracefulness of some special class of flower petal graphs.

On local antimagic chromatic number of cycle-related join graphs

An edge labeling of a connected graph $G = (V, E)$ is said to be local antimagic if it is a bijection $f:E \to\{1,\ldots ,|E|\}$ such that for any pair of adjacent vertices $x$ and $y$, $f^+(x)\not= f^+(y)$, where the induced vertex label $f^+(x)= \sum f(e)$, with $e$ ranging over all the edges incident to $x$. The local antimagic chromatic number of $G$, denoted by $\chi_{la}(G)$, is the minimum number of distinct induced vertex labels over all local antimagic labelings of $G$. In this paper, several sufficient conditions for $\chi_{la}(H)\le \chi_{la}(G)$ are obtained, where $H$ is obtained from $G$ with a certain edge deleted or added. We then determined the exact value of the local antimagic chromatic number of many cycle related join graphs.

Affirmative Solutions on Local Antimagic Chromatic Number

An edge labeling of a connected graph $$G = (V, E)$$ is said to be local antimagic if it is a bijection $$f:E \rightarrow \{1,\ldots ,|E|\}$$ such that for any pair of adjacent vertices x and y, $$f^+(x)\not = f^+(y)$$ , where the induced vertex label $$f^+(x)= \sum f(e)$$ , with e ranging over all the edges incident to x. The local antimagic chromatic number of G, denoted by $$\chi _{la}(G)$$ , is the minimum number of distinct induced vertex labels over all local antimagic labelings of G. In this paper, we give counterexamples to the lower bound of $$\chi _{la}(G \vee O_2)$$ that was obtained in [Local antimagic vertex coloring of a graph, Graphs Combin. 33:275–285 (2017)]. A sharp lower bound of $$\chi _{la}(G\vee O_n)$$ and sufficient conditions for the given lower bound to be attained are obtained. Moreover, we settled Theorem 2.15 and solved Problem 3.3 in the affirmative. We also completely determined the local antimagic chromatic number of complete bipartite graphs.

Matrix Coding Technique on Sunflower Graphs with Edge Product Cordial Labeling

In this paper we develop a technique of coding a secret messages using Sun Flower graphs SFn by subdividing edges and applying edge product cordial labeling. The induced vertex labeling function defined by the product of labels of incident edges to each vertex is such that the number of edges with label 0 and 1 differ by atmost 1. Here, we discuss edge product cordial labeling for sunflower graphs subdivided edges and developed a coding technique of transforming plain text (Text message) in to Cipher Text (Matrix code) and established algorithm. By recent development in coding theory involving programming concepts in any computer languages, this article will revisit without programming method, the matrix coding technique is developed.

Super vertex mean labeling of cycles through different ways

A super vertex mean labeling 𝑓 of a (p,q) - graph G = (V,E) is defined as an injection from E to the set {1,2,3,···,p+q} that induces for each vertex v the label defined by the rule , where Ev denotes the set of edges in G that are incident at the vertex v, such that the set of all edge labels and the induced vertex labels is {1,2,3,···,p+q}. In this paper, we investigate the super vertex mean labeling behavior of cycles by giving various ways by which they can be labeled.

A note on V4 magic labelings of graphs

For any abelian group (A, +) with the identity element 0, we denote A* = A\\{0}. Let G be a graph with a vertex set V(G) and an edge set E(G). Any mapping is called an edge labeling. If is an edge labeling, then we can define an induced vertex labeling as follows:A graph G is known as A-magic if there exists a labeling such that the induced vertex labeling is a constant map. That is, there exists an element k ∈ A such that for all . If k = 0, we say that the graph G is zero sum A-magic, otherwise G is k-sum A-magic. Observe that there exist graphs which are both zero sum and k-sum A magic. The Klein 4 group V4 is the direct sum ℤ2 ⊕ ℤ2 = {(0, 0), (1, 0), (0, 1), (1, 1)}. For simplicity we denote (0, 0), (0, 1), (1, 0), (1, 1) by 0, a, b, c respectively. In this paper we prove that the graph , the spanning subgraph of Kn (n ≥ 4) with = E(Kn) \\ {vivj :1 ≤ i < j ≤ r}, 2 ≤ r ≤ n − 2 is a-sum V4 magic if and only if n is even and zero sum V4 magic for all n. We also investigated a-sum V4 magic labeling as well as the zero sum V4 magic labeling of . We construct a new graph from by attaching r edges at the first r vertices v1, v2, … , vr and n – r triangles at the remaining vertices of . Finally, we prove that is a-sum V magic if and only if n and r are of the same parity.

Group-antimagic Labelings of Multi-cyclic Graphs

Let A be a non-trivial abelian group. A connected simple graph G = (V,E) is A-antimagic if there exists an edge labeling f : E(G) → A\{0} such that the induced vertex labeling f+ : V (G) → A, defined by f+(v) = Σ {f(u, v) : (u, v) ∈ E(G)}, is a one-to-one map. The integer-antimagic spectrum of a graph G is the set IAM(G) = {k : G is Zk-antimagic and k ≥ 2}. In this paper, we analyze the integer-antimagic spectra for various classes of multi-cyclic graphs.