Antimagic Labeling Research Articles

Shifted-Antimagic Labelings for Graphs

The concept of antimagic labelings of a graph is to produce distinct vertex sums by labeling edges through consecutive numbers starting from one. A long-standing conjecture is that every connected graph, except a single edge, is antimagic. Some graphs are known to be antimagic, but little has been known about sparse graphs, not even trees. This paper studies a weak version called k-shifted-antimagic labelings which allow the consecutive numbers starting from \(k+1\), instead of starting from 1, where k can be any integer. This paper establishes connections among various concepts proposed in the literature of antimagic labelings and extends previous results in three aspects: Some classes of graphs, including trees and graphs whose vertices are of odd degrees, which have not been verified to be antimagic are shown to be k-shifted-antimagic for sufficiently large k. Some graphs are proved k-shifted-antimagic for all k, while some are proved not for some particular k. Disconnected graphs are also considered.

On rainbow antimagic coloring of graphs

A rainbow antimagic coloring is one of new topics in graph theory. This topic is an expansion of rainbow coloring that is combined with antimagic labeling. The graphs are labeled with an antimagic labeling, and then the sum of vertex label have to obtain a rainbow coloring. The aim of the rainbow antimagic coloring research is to find the minimum number of color, called rainbow antimagic connection number, denoted by rac(G). In this research, we studied some simple graphs to be colored with rainbow antimagic coloring. The graphs we used are lollipop, stacked book, Dutch windmill, flowerpot and dragonfly. In this research, we aimed to develop new theorem. Based on the results, we got some theorems about rac(G) for the rainbow antimagic coloring. In addition, on some graphs that we use, we get rac(G) equal to rc(G), rainbow connection number. Instead, on the others, the value of rac(G) cannot reach the rc(G).

Combinatorial Configurations in the Definition of Antimagic Labelings of Graphs

We have formalized the definition of graph labeling in terms of combinatorial configurations. We have investigated the connection between edge and vertex (a, d)–distance antimagic labelings with such well-known configurations as separating systems and magic rectangle sets. We have obtained a solution to the problem of construction of indicated labelings for some types of graphs and certain values of a and d.

On the local antimagic labeling of graphs amalgamation

Let G be a simple connected graph, an ordered pair of sets G(V, E), with V is a set of vertices and E is a set of edges. Graph coloring has been one of the most popular branches in topics of graph theory. In 2017 Arumugam et al. developed a new notion of coloring, namely local antimagic coloring of a graph. This concept is a combination of graph labeling and graph coloring. The local antimagic of graphs is one of the colorings of graph theory that is interesting to study. By the definition, the local antimagic coloring is a bijection f : E(G) ƒ> {1, 2, 3,…, |E(G)|} if for any two adjacent vertices u and v, w(u) = w(v), where w(v) = ΣeeE(v) f (e), and E(u) is the set of edges incident to u. Thus any local antimagic labeling induces a proper vertex coloring of G where the vertex v is assigned the color w(v). The local antimagic chromatic number xla(G) is the minimum number of colors taken over all colorings induced by local antimagic labelings of G. In this paper, we will study the local antimagic coloring of amalgamation of graphs.

On rainbow antimagic coloring of special graphs

Let G(V, E) be a connected, undirected and simple graph with vertex set V(G) and edge set E(G). A labeling of a graph G is a bijection f from V(G) to the set {1, 2,…, | V(G)|}. The bijection f is called rainbow antimagic vertex labeling if for any two edge uv and u’v’ in path x — y,w(uv) = w(u’v’) w(uv), where w(uv) = f (u) + f (v) and x,y ∈ V(G). A graph G is a rainbow antimagic connection if G has a rainbow antimagic labeling. Thus any rainbow antimagic labeling induces a rainbow coloring of G where the edge uv is assigned with the color w(uv). The rainbow antimagic connection number of G, denoted by rac(G), is the smallest number of colors taken over all rainbow colorings induced by rainbow antimagic labeling of G. In this paper, we show the exact value of the rainbow antimagic connection number of jahangir graph J2,m, lemon graph Lem, firecracker graph (Fm,3), complete bipartite graph (K2,m), and double star graph (Sm,m).

Caterpillars are Antimagic

An antimagic labeling of a graph G is a bijection from the set of edges E(G) to $$\{1,2,\ldots ,|E(G)|\}$$ , such that all vertex sums are pairwise distinct, where the vertex sum at vertex u is the sum of the labels assigned to the edges incident to u. A graph is called antimagic when it has an antimagic labeling. Hartsfield and Ringel conjectured that every simple connected graph other than $$K_2$$ is antimagic and the conjecture remains open even for trees. Here, we prove that caterpillars are antimagic by means of an $$O(n \log n)$$ algorithm.

A network communication through McGee graph and Antimagic labeling

A network communication through McGee graph and Antimagic labeling

(Super)(a, d) - H- Antimagic total labeling of super sub division of cycle

(Super)(a, d) - H- Antimagic total labeling of super sub division of cycle

On (a,d)-antimagic labelings of Hn, FLn and mCn

In this paper, we derive the necessary condition for an (a,d )- antimagic labeling of some new classes of graphs such as <em>H</em><em><sub>n</sub>, F L<sub>n</sub> </em>and <em>mC</em><em><sub>n</sub></em>. We prove that <em>H</em><em><sub>n</sub> </em>is (7<em>n </em>+2<em>, </em>1)-antimagic and <em>mC</em><em><sub>n</sub> </em>is ((mn+3)/2,1)- antimagic. Also we prove that <em>F L</em><em><sub>n</sub> </em>has no ((n+1)/2,4)- antimagic labeling.

Group Labeling of some Graphs

Group labelings of magic and antimagic types are analyzed and relationship between them for the graph and its complement are established. The concept of closed group distance magic labeling is introduced. The existence conditions for $$ {Z}_2^{2m} $$ -distance magic labeling of the graph $$ {C}_4^m $$ are found, and a method for its construction is proposed. The existence conditions for $$ {Z}_2^r $$ -distance magic and antimagic labelings of the Cartesian product of regular graphs are established. The results of group remote magic labeling of the connection of two graphs are obtained.

Distance antimagic labelings of Cartesian product of graphs

Let be a graph of order n. Let be a bijection. The weight w(v) of a vertex v with respect to the labeling f is defined by where N(v) is the open neighborhood of v. The labeling f is called a distance antimagic labeling if for any two distinct vertices v1, v2 in V. In this paper we investigate the existence of distance antimagic labeling for the Cartesian product where the graphs G and H are cycles or complete graphs.

Antimagic orientations of graphs with large maximum degree

Given a digraph D with m arcs, a bijection τ:A(D)→{1,2,…,m} is an antimagic labeling of D if no two vertices in D have the same vertex-sum, where the vertex-sum of a vertex u in D under τ is the sum of labels of all arcs entering u minus the sum of labels of all arcs leaving u. We say (D,τ) is an antimagic orientation of a graph G if D is an orientation of G and τ is an antimagic labeling of D. Motivated by the conjecture of Hartsfield and Ringel (1990) on antimagic labelings of graphs, Hefetz et al. (2010) initiated the study of antimagic orientations of graphs, and conjectured that every connected graph admits an antimagic orientation. This conjecture seems hard, and few related results are known. However, it has been verified to be true for regular graphs and biregular bipartite graphs. In this paper, we prove that every connected graph G on n≥9 vertices with maximum degree at least n−5 admits an antimagic orientation.

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.

Encrypting and decrypting algorithms using strong face graph of a tree

The encryption and decryption is based upon the type of cryptographic scheme being employed and also in some form of key. It is most closely associated with the development and creation of the mathematical algorithms used to encrypt and decrypt messages. The combination of graph labelling techniques together with cryptography to encrypt and decrypt the numbers has been already an ongoing aspect of research. In this paper, the concept of face antimagic labelling is used for a strong face of duplication of all vertices by the edges of a tree to encrypt and decrypt 13 secret numbers.

Local antimagic chromatic number of trees - I

Let G = (V, E) be a connected graph with |V|= n and |E|= m. A bijection f : E → {1, 2, … , m} is called a local antimagic lableing if for any two adjacent vertices u and v, w(u) ≠ w(v), where w(u)= ∑ e ϵE(u) f (e) and E(u) is the set of edges incident to u. Thus any local antimagic labeling induces a proper vertex coloring of G where the vertex v is assigned the color w(v). The local antimagic chromatic number χla (G) is the minimum number of colors taken over all colorings induced by local antimagic labelings of G. In this paper we determine the local antimagic chromatic number of several families of trees.

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.

Ladders and fan graphs are cycle-antimagic

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 $f:V\cup E\to\{1,2,\cdots,|V|+|E|\}$ such that, for all subgraphs $H'$ of $G$ isomorphic to $H$, the $H'$-weights, $wt_f(H')= \sum_{v\in V(H')} f(v) + \sum_{e\in E(H')} f(e)$, form an~arithmetic progression with the initial term $a$ and the common difference $d$. Such a labeling is called {\it super} if the smallest possible labels appear on the vertices. In this paper we prove the existence of super $(a,d)$-$H$-antimagic labelings of fan graphs and ladders for $H$ isomorphic to a cycle.

The analysis of the implementation of project based learning and its influence to the student deductive reasoning based on cognitive style on solving super edge local antimagic total labelling

In this study, we tried to test students’ deductive reasoning based on cognitive style in the application of super edge total local antimagic labels by applying Project Based Learning (PjBL) learning models. This study uses a mixed method, namely qualitative methods and quantitative methods. The research subjects consisted of two classes, namely 28 control class students and 30 experimental class students. The instruments used in this study were pretest, posttest, observation sheet, and interview sheet. Before the research activities, we tested the homogeneity of both classes using the results of the pretest. Homogeneity test results show the sig value is 0.481> 0.05, so the average difference between the two classes is not significant. This shows that both classes are homogeneous or have the same ability. Next, we also discuss the normality of using posttest data. This shows that the independent sample test on the posttest data from the control class and the experimental class obtained a sig score of 0.657 <0.05, so that it was significant which meant the difference between the control class and the experimental class. While the effect of punishment increases based on students’ cognitive style, it shows that 52% are in the very high category, 31% depend on the high category, 13% depend on the average category, and 4% depend on the low category. Project-Based Learning Model (PjBL) and the influence of deductive reasoning based on students’ cognitive styles in the super edge study of total local antimagic labeling.

Some results on (a,d)-distance antimagic labeling

Let G = (V,E) be a graph of order N and f : V → {1, 2,...,N} be a bijection. For every vertex v of graph G, we define its weight w(v) as the sum ∑u∈N(v) f(u), where N(v) denotes the open neighborhood of v. If the set of all vertex weights forms an arithmetic progression {a, a + d, a + 2d, . . . , a + (N − 1)d}, then f is called an (a, d)-distance antimagic labeling and the graph G is called (a, d)-distance antimagic graph. In this paper we prove the existence or non-existence of (a, d)- distance antimagic labeling of some well-known graphs.

Local antimagic orientation of graphs

An antimagic labelling of a digraph D with m arcs is a bijection from the set of arcs of D to $$\{1,\ldots ,m\}$$ such that any two vertices have distinct vertex-sums, where the vertex-sum of a vertex $$v\in V(D)$$ is the sum of labels of all arcs entering v minus the sum of labels of all arcs leaving v. An orientation D of a graph G is antimagic if D has an antimagic labelling. In 2010, Hefetz, M$$\ddot{\text {u}}$$tze and Schwartz conjectured that every connected graph admits an antimagic orientation. The conjecture is still open, even for trees. Motivated by directed version of the well-known 1-2-3 Conjecture, we deal with vertex-sums such that only adjacent vertices must be distinguished. An orientation D of a graph G is local antimagic if there is a bijection from E(G) to $$\{1,\ldots ,|E(G)|\}$$ such that any two adjacent vertices have distinct vertex-sums. We prove that every graph with maximum degree at most 4 admits a local antimagic orientation by Alon’s Combinatorial Nullstellensatz.