Antimagic Labeling Research Articles

ON RAINBOW ANTIMAGIC COLORING OF SNAIL GRAPH(S_n ), COCONUT ROOT GRAPH (Cr_(n,m) ), FAN STALK GRAPH (Kt_n ) AND THE LOTUS GRAPH(Lo_n )

Rainbow antimagic coloring is a combination of antimagic labeling and rainbow coloring. Antimagic labeling is labeling of each vertex of the graph with a different label, so that each the sum of the vertices in the graph has a different weight. Rainbow coloring is part of the rainbow-connected edge coloring, where each graph has a rainbow path. A rainbow path in a graph is formed if two vertices on the graph do not have the same color. If the given color on each edge is different, for example in the function it is colored with a weight , it is called rainbow antimagic coloring. Rainbow antimagic coloring has a condition that every two vertices on a graph cannot have the same rainbow path. The minimum number of colors from rainbow antimagic coloring is called the rainbow antimagic connection number, denoted by In this study, we analyze the rainbow antimagic connection number of snail graph , coconut root graph , fan stalk graph and lotus graph .

Antimagic orientation of forests

An antimagic labeling of a digraph D with n vertices and m arcs is a bijection from the set of arcs of D to {1,2,…,m} such that all n oriented vertex-sums are pairwise distinct, where the oriented vertex-sum of a vertex is the sum of labels of all arcs entering that vertex minus the sum of labels of all arcs leaving it. A graph G admits an antimagic orientation if G has an orientation D such that D has an antimagic labeling. Hefetz, Mütze and Schwartz conjectured every connected graph admits an antimagic orientation. In this paper, we support this conjecture by proving that any forest obtained from a given forest with at most one isolated vertex by subdividing each edge at least once admits an antimagic orientation.

Antimagic Labeling of Forests

An antimagic labeling of a graph G(V,E) is a bijection f mapping from E to the set {1,2,…, |E|}, so that for any two different vertices u and v, the sum of f(e) over all edges e incident to u, and the sum of f(e) over all edges e incident to v, are distinct. We call G antimagic if it admits an antimagic labeling. A forest is a graph without cycles; equivalently, every component of a forest is a tree.
 It was proved by Kaplan, Lev, and Roditty in 2009, and by Liang, Wong, and Zhu in 2014 that every tree with at most one vertex of degree two is antimagic. A major tool used in the proof is the zero-sum partition introduced by Kaplan, Lev, and Roditty in 2009. In this article, we provide an algorithmic representation for the zero-sum partition method and apply this method to show that every forest with at most one vertex of degree two is also antimagic.

Graph antimagic labeling: A survey

An antimagic labeling of a simple graph [Formula: see text] is a bijection [Formula: see text] such that [Formula: see text] for any two vertices [Formula: see text] in [Formula: see text]. We survey the results about antimagic labelings and other labelings motivated by antimagic labelings of graphs, and present some conjectures and open questions.

Local total anti-magic chromatic number of graphs

Let G=(V,E) be a graph without isolated vertices and let |V(G)|=n and |E(G)|=m. A bijection π:V(G)∪E(G)→{1,2,....,n+m} is said to be local total anti-magic labeling of a graph G if it satisfies the conditions: (i.) for any edge uv, ω(u)≠ω(v), where u and v in V(G) (ii.) for any two adjacent edges e and e′, ω(e)≠ω(e′) (iii.) for any edge uv∈E(G) is incident to the vertex v, ω(v)≠ω(uv), where weight of vertex u is, ω(u)=∑e∈S(u)π(e), S(u) is the set of edges with every edge of S(u) one end vertex is u and an edge weight is ω(e=uv)=π(u)+π(v). In this paper, we have introduced a local total anti-magic labeling (LTAL) and the local total anti-magic chromatic number (LTACN). Also, we obtain the LTACN for the graphs Pn, K1,n, Fn and Sn,n.

Local edge (a, d) –antimagic coloring on sunflower, umbrella graph and its application

Suppose a graph G = (V, E) is a simple, connected and finite graph with vertex set V(G) and an edge set E(G). The local edge antimagic coloring is a combination of local antimagic labelling and edge coloring. A mapping f∶ V (G)→ {1, 2, ..., |V (G)|} is called local edge antimagic coloring if every two incident edges e_1and e_2, then the edge weights of e_1and e_2 maynot be the same, w(e_1) ≠ w(e_2), with e = uv ∈ G, w(e) = f(u)+ f(v) with the rule that the edges e are colored according to their weights, w_e. Local edge antimagic coloring has developed into local (a,d)-antimagic coloring. Local antimagic coloring is called local (a,d)-antimagic coloring if the set of edge weights forms an arithmetic sequence with a as an initial value and d as a difference value. The graphs used in this study are sunflower graphs and umbrella graphs. This research will also discuss one of the applications of local edge (a,d)-antimagic coloring, namely the design of the Sidoarjo line batik motif. The result show that χ_le(3,1) (Sf_n) = 3n and χ_le(3n/2,1) (U_(m,n) ) = m+1 . The local (a,d)-antimagic coloring is formed into a batik motif design with characteristics from the Sidoarjo region.

Generation of anti-magic graphs from binary graph products

An anti-magic labeling of a graph G is a one-to-one correspondence between E(G) and {1, 2, ··· , |E|} such that the vertex-sum for distinct vertices are different. Vertex-sum of a vertex u ∈ V (G) is the sum of labels assigned to edges incident to the vertex u. It was conjectured by Hartsfield and Ringel that every tree other than K2 has an anti-magic labeling. In this paper, we consider various binary graph products such as corona, edge corona and rooted products to generate anti-magic graphs. We prove that corona products of an anti-magic regular graph G with K1 and K2 are anti-magic. Further, we prove that rooted product of two anti-magic trees are anti-magic. Also, we prove that rooted product of an anti-magic graph with an anti-magic tree admits anti-magic labeling.

On the study of Rainbow Antimagic Coloring of Special Graphs

Let be a connected graph with vertex set and edge set . The bijective function is said to be a labeling of graph where is the associated weight for edge . If every edge has different weight, the function is called an edge antimagic vertex labeling. A path in the vertex-labeled graph , with every two edges satisfies is said to be a rainbow path. The function is called a rainbow antimagic labeling of , if for every two vertices , there exists a rainbow path. Graph admits the rainbow antimagic coloring, if we assign each edge with the color of the edge weight . The smallest number of colors induced from all edge weights of edge antimagic vertex labeling is called a rainbow antimagic connection number of , denoted by . In this paper, we study rainbow antimagic connection numbers of octopus graph , sandat graph , sun flower graph , volcano graph and semi jahangir graph Jn.

On Rainbow Antimagic Coloring of Joint Product of Graphs

Let be a connected graph with vertex set and edge set . A bijection from to the set is a labeling of graph . The bijection is called rainbow antimagic vertex labeling if for any two edge and in path , where and . Rainbow antimagic coloring is a graph which has a rainbow antimagic labeling. Thus, every rainbow antimagic labeling induces a rainbow coloring G where the edge weight is the color of the edge . The rainbow antimagic connection number of graph is the smallest number of colors of all rainbow antimagic colorings of graph , denoted by . In this study, we studied rainbow antimagic coloring and have an exact value of rainbow antimagic connection number of joint product of graph where is graph , graph , graph , graph and graph .

Local distance antimagic chromatic number for the union of star and double star graphs

UDC 519.17 Let G = ( V , E ) be a graph on p vertices with no isolated vertices. A bijection f from V to { 1,2 ,3 , … , p } is called a local distance antimagic labeling if, for any two adjacent vertices u and v , we receive distinct weights (colors), where a vertex x has the weight w ( x ) = ∑ v ϵ N ( x ) f ( v ) . The local distance antimagic chromatic number χ l ⅆ a ( G ) is defined as the least number of colors used in any local distance antimagic labeling of G . We determine the local distance antimagic chromatic number for the disjoint union of t copies of stars and double stars.

Distance antimagic labelings of product graphs

A graph G is distance antimagic if there is a bijection f : V ( G )→{1, 2, …, | V ( G )|} such that for every pair of distinct vertices x and y applies w ( x )≠ w ( y ) , where w ( x )= Σ z ∈ N ( x ) f ( z ) and N ( x ) is the neighbourhood of x , i.e., the set of all vertices adjacent to x . It was conjectured that a graph is distance antimagic if and only if each vertex in the graph has a distinct neighbourhood. In this paper, we study the truth of the conjecture by posing sufficient conditions and constructing distance antimagic product graphs; the products under consideration are join, corona, and Cartesian.

On regular d-handicap tournaments

A k-regular d-handicap tournament is an incomplete tournament in which n teams, ranked according to the natural numbers, play exactly k < n − 1 different teams exactly once and the strength of schedule of the i t h ranked team is d more than the ( i − 1) s t ranked team for some d ≥ 1 . That is, strength of schedules increase arithmetically by d with strength of team. A d-handicap distance antimagic labeling of a graph G = ( V , E ) of order n is a bijection ℓ : V → {1,2,…, n } with induced weight function w ( x i )= Σ x j ∈ N ( x i ) l ( x j ) such that ℓ( x i )= i and the sequence of weights w ( x 1 ), w ( x 2 ),…, w ( x n ) forms an arithmetic sequence with difference d ≥ 1. A graph G which admits such a labeling is called a d-handicap graph . Constructing a k -regular d -handicap tournament on n teams is equivalent to finding a k -regular d -handicap graph of order n . For d = 1 and n even, the existence has recently been completely settled for all pairs ( n , k ) , and some results are known for d = 2 . For d > 2, the only known result is restricted to the case where n is divisible by 2 d + 2 . In this paper, we construct infinite families of d -handicap graphs where the order is not restricted to a power of 2.

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.

Pelabelan Jarak Tak Teratur Titik Pada Graf Persahabatan Lengkap Diperumum

Graph labeling is the labeling of graph elements such as vertex, edge and both. distance vertex irregular labeling is a type of labeling resulting from the development of distance magic labeling and (a, b)-distance anti-magic labeling. Let , be a simple graph. The distance vertex irregular labeling of is a vertex labeling so that the weight of each vertex is different. The weight of is calculated based on the sum of vertices label in the set of neighboring vertex , namely Distance vertex irregularity strength of , denoted as d , is the smallest value of the largest label so that has a distance vertex irregular labeling. This study aims to construct a generalized complete friendship graph , determine the labeling function, determine the distance vertex irregularity strength then formulate and prove the theorem resulting from the labeling. The object of this research is to label each vertex on a generalized complete friendship graph. This research method is a literature study obtained through various sources. Based on the research results, it is known that the graph has distance vertex irregular labeling. For an integer m and n, , the labeling function of is and . Distance vertex irregularity strength of generalized complete friendship graph is .

Antimagic labeling for unions of graphs with many three-paths

Let G be a graph with m edges and let f be a bijection from E(G) to {1,2,…,m}. For any vertex v, denote by ϕf(v) the sum of f(e) over all edges e incident to v. If ϕf(v)≠ϕf(u) holds for any two distinct vertices u and v, then f is called an antimagic labeling of G. We call G antimagic if such a labeling exists. Hartsfield and Ringel [10] conjectured that all connected graphs except P2 are antimagic. Denote the disjoint union of graphs G and H by G∪H, and the disjoint union of t copies of G by tG. For an antimagic graph G (connected or disconnected), we define the parameter τ(G) to be the maximum integer such that G∪tP3 is antimagic for all t⩽τ(G). Chang, Chen, Li, and Pan showed that for all antimagic graphs G, τ(G) is finite [3]. Further, Shang, Lin, Liaw [19] and Li [14] found the exact value of τ(G) for special families of graphs: star forests and balanced double stars, respectively. They did this by finding explicit antimagic labelings of G∪tP3 and proving a tight upper bound on τ(G) for these special families. In the present paper, we generalize their results by proving an upper bound on τ(G) for all graphs. For star forests and balanced double stars, this general bound is equivalent to the bounds given in [19] and [14] and tight. In addition, we prove that the general bound is also tight for every other graph we have studied, including an infinite family of jellyfish graphs, cycles Cn with 3⩽n⩽9, and the double triangle 2C3.

On join product and local antimagic chromatic number of regular graphs

Let $$G = (V,E)$$ be a connected simple graph of order p and size q. A graph G is called local antimagic if G admits a local antimagic labeling. A bijection $$f \colon E \to \{1,2,\ldots,q\}$$ is called a local antimagic labeling of G if for any two adjacent vertices $$u$$ and $$v$$ , we have $$f^+(u) \ne f^+(v)$$ , where $$f^+(u) = \sum_{e\in E(u)} f(e)$$ , and $$E(u)$$ is the set of edges incident to $$u$$ . Thus, any local antimagic labeling induces a proper vertex coloring of G if vertex $$v$$ is assigned the color $$f^+(v)$$ . The local antimagic chromatic number, denoted $$\chi_{la}(G)$$ , is the minimum number of induced colors taken over local antimagic labeling of G. Let G and H be two vertex disjoint graphs. The join graph of G and H, denoted $$G \vee H$$ , is the graph with $$V(G\vee H) = V(G) \cup V(H)$$ and $$E(G\vee H) = E(G) \cup E(H) \cup \{uv \mid u\in V(G)$$ , $$v \in V(H)\}$$ . In this paper, we investigated $$\chi_{la}(G\vee H)$$ . Consequently, we show the existence of non-complete regular graphs with arbitrarily large order, regularity and local antimagic chromatic numbers.

On the study of Rainbow Antimagic Connection Number of Corona Product of Graphs

Given that a graph G = (V, E). By an edge-antimagic vertex labeling of graph, we mean assigning labels on each vertex under the label function f : V → {1, 2, . . . , |V (G)|} such that the associated weight of an edge uv ∈ E(G), namely w(xy) = f(x) + f(y), has distinct weight. A path P in the vertex-labeled graph G is said to be a rainbow path if for every two edges xy, x′y ′ ∈ E(P) satisfies w(xy) ̸= w(x ′y ′ ). The function f is called a rainbow antimagic labeling of G if for every two vertices x and y of G, there exists a rainbow x − y path. When we assign each edge xy with the color of the edge weight w(xy), thus we say the graph G admits a rainbow antimagic coloring. The rainbow antimagic connection number of G, denoted by rac(G), is the smallest number of colors induced from all edge weight of antimagic labeling. In this paper, we will study the rac(G) of the corona product of graphs. By the corona product of graphs G and H, denoted by G ⊙ H, we mean a graph obtained by taking a copy of graph G and n copies of graph H, namely H1, H2, ..., Hn, then connecting vertex vi from the copy of graph G to every vertex on graph Hi , i = 1, 2, 3, . . . , n. In this paper, we show the exact value of the rainbow antimagic connection number of Tn ⊙ Sm where Tn ∈ {Pn, Sn, Sn,p, Fn,3}.

On Antimagic Labeling for Some Families of Graphs

Antimagic labeling of a graph with vertices and edges is assigned the labels for its edges by some integers from the set , such that no two edges received the same label, and the weights of vertices of a graph are pairwise distinct. Where the vertex-weights of a vertex under this labeling is the sum of labels of all edges incident to this vertex, in this paper, we deal with the problem of finding vertex antimagic edge labeling for some special families of graphs called strong face graphs. We prove that vertex antimagic, edge labeling for strong face ladder graph , strong face wheel graph , strong face fan graph , strong face prism graph and finally strong face friendship graph .

On local antimagic total labeling of complete graphs amalgamation

Let \(G = (V,E)\) be a connected simple graph of order \(p\) and size \(q\). A graph \(G\) is called local antimagic (total) if \(G\) admits a local antimagic (total) labeling. A bijection \(g : E \to \{1,2,\ldots,q\}\) is called a local antimagic labeling of $ if for any two adjacent vertices \(u\) and \(v\), we have \(g^+(u) \ne g^+(v)\), where \(g^+(u) = \sum_{e\in E(u)} g(e)\), and \(E(u)\) is the set of edges incident to \(u\). Similarly, a bijection \(f:V(G)\cup E(G)\to \{1,2,\ldots,p+q\}\) is called a local antimagic total labeling of \(G\) if for any two adjacent vertices \(u\) and \(v\), we have \(w_f(u)\ne w_f(v)\), where \(w_f(u) = f(u) + \sum_{e\in E(u)} f(e)\). Thus, any local antimagic (total) labeling induces a proper vertex coloring of \(G\) if vertex \(v\) is assigned the color \(g^+(v)\) (respectively, \(w_f(u)\)). The local antimagic (total) chromatic number, denoted \(\chi_{la}(G)\) (respectively \(\chi_{lat}(G)\)), is the minimum number of induced colors taken over local antimagic (total) labeling of \(G\). In this paper, we determined \(\chi_{lat}(G)\) where \(G\) is the amalgamation ofcomplete graphs. Consequently, we also obtained the local antimagic (total) chromatic number of the disjoint union of complete graphs, and the join of \(K_1\) and amalgamation of complete graphs under various conditions. An application of local antimagic total chromatic number is also given.

Encryption and decryption of a word into weighted graph using super-edge anti-magic total labeling of Bi-star graph

Graph labeling is an important topic in Graph theory, a branch of Mathematics, having applications in the field of Networks, radio astronomy and cryptography etc. In most of encryption algorithms, words are encrypted as words. But our implementation is based on the algorithm that encrypts words into numbers, with the help of an edge weighted graph. Mapping of certain positive integers to graph elements in such a way that weights of all edges computed as, sum of the labels of all graph elements, are distinct is termed as Antimagic labeling In this paper, we present an algorithm that encrypts words into edge weighted graphs using super-edge antimagic total labeling of Bi-star graph and illustrate with an example.