Antimagic 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.

Graceful Antimagic Path and Cycle Related Graphs

A graph with vertices and edges is said to be antimagic if its edges can be labeled with such that the weights of vertices of are pairwise distinct. The graceful labeling of a graph with edges is an assignment of integers from the set to the vertices of , such that no two vertices receive the same label, where each edge is assigned the absolute value of the difference between the labels of its end vertices and the resulting edge labeling runs from to is inclusive. Moreover, if the induced edge labeling is simultaneously antimagic, that is, the sums of the labels of all edges incident to a given vertex are pairwise distinct for different vertices, we call the graceful labeling graceful antimagic. In this study, we will exhibit the existence of graceful antimagic labeling for two families of graphs the first derived from the path graph Pn, and the second from the cycle graph Cn. Both families were derived using the idea of a rooted product between the two graphs.

Palindromic Antimagic Labeling of Products of Paw and Banner Graph

This article presents a novel classification of Antimagic labeling referred to as Palindromic antimagic. Palindromic Antimagic labeling pertains to the assignment of palindromic numbers {℘1, ℘2, ℘3 . . . ℘q} to the set of edges of a graph G = (V, E), where G consists of p vertices and q edges. This labeling is characterized by being invertible, meaning that each edge is uniquely associated with a palindromic number. Additionally, it involves an injective mapping of vertex labeling, where the sum of incident edges for any given vertex is distinct from one another. Inthis study, we examine the palindromic antimagic labeling of Cartesian and Tensor product of certain Tadpole graphs, such as the paw and banner graph.

Local distance antimagic cromatic number of join product of graphs with cycles or paths

Let $G$ be a graph of order $p$ without isolated vertices. A bijection $f: V \to \{1,2,3,\dots,p\}$ is called a local distance antimagic labeling, if $w_f(u)\ne w_f(v)$ for every edge $uv$ of $G$, where $w_f(u)=\sum_{x\epsilon N(u)} {f(x)}$. The local distance antimagic chromatic number $\chi_{lda}(G)$ is defined to be the minimum number of colors taken over all colorings of $G$ induced by local distance antimagic labelings of $G$. In this paper, we determined the local distance antimagic chromatic number of some cycles, paths, disjoint union of 3-paths. We also determined the local distance antimagic chromatic number of join products of some graphs with cycles or paths.

A new class of disconnected antimagic graphs

A graph G is antimagic if there exists a bijection f from E(G) to {1, 2, . . . , |E(G)|} such that the vertex sums for all vertices of G are distinct, where the vertex sum is defined as the sum of the labels of all incident edges. Hartsfield and Ringel conjectured that every connected graph other than K_2 admits an antimagic labeling. It is still a challenging problem to address antimagicness in the case of disconnected graphs. In this paper, we study antimagicness for the disconnected graph that is constructed as the direct product of a star and a path.

Antimagic Labeling for Some Snake Graphs

A graph with q edges is called antimagic if its edges can be labeled with 1, 2, 3, ..., q without repetition such that the sums of the labels of the edges incident to each vertex are distinct. In this paper we study antimagic labeling of double triangular snake, alternate triangular snake, double alternate triangular snake, quadrilateral snake, double quadrilateral snake, alternate quadrilateral snake, double alternate quadrilateral snake.

On local distance antimagic chromatic number of graphs disjoint union with 1-regular graphs

On local distance antimagic chromatic number of graphs disjoint union with 1-regular graphs

Super (a,d)-P_2⨀P_m-Antimagic Total Labeling of Corona Product of Two Paths

Graph labeling involves mapping the elements of a graph (edges and vertices) to a set of positive integers. The concept of an anti-magic super outer labeling (a,d)-H pertains to assigning labels to the vertices and edges of a graph using natural numbers {1,2,3,...,p+q}. The weights of the outer labels H form an arithmetic sequence {a,a+d,a+2d,...,a+(k-1)d}, where 'a' represents the first term, 'd' is the common difference, and 'k' denotes the total number of outer labels, with the smallest label assigned to a vertex. This study investigates the super (a,d)-P_2⨀P_m-antimagic total labeling of the corona product P_n⨀P_m, where n and m are both greater than or equal to 3. We define the labeling functions for vertices and edges based on the partitioning of labels into three subsets. Using k-balanced and (k,δ)-anti balanced multisets, we demonstrate that for m being odd, P_n⨀P_m is super (9m^2 n+4mn+m-n+3,1)-P_2 ⨀▒P_(m ) -antimagic, and for m being even, P_n⨀P_m is super (9m^2 n+4mn+m-2n+5,3)-P_2 ⨀▒P_(m ) -antimagic. The labeling scheme is illustrated through examples. For the case when m is odd, an antimagic total labeling of P_3 ⨀▒P_3 forms a super (282,1)- P_2 ⨀▒P_(3 ) -antimagic labeling. In the case of even m, an antimagic total labeling of P_3 ⨀▒P_(4 ) results in a super (483,3)- P_2 ⨀▒P_(4 ) -antimagic labeling. Both of these examples provide insights into the antimagic properties of corona products.

Bi-Magic and Anti-Magic Labeling on M-Fuzzy Graph

The conceptions of M-fuzzy anti-magic and M-fuzzy bi-magic labeling are developed by the proposed algorithms. M-fuzzy anti-magic labeling for standard graphs such as star and path graph explained. M-fuzzy bi-magic labeling for a few graphs such as star and bi-star is determined. Also, we explored the characteristics of such labeling applied to these graphs.

Antimagic labeling in butterfly and benes networks

This paper focus on the dynamic and rapidly evolving field of Graph Labeling research. In this paper, Antimagic Labeling for Benes and Butterfly networks has been determined. The findings can be utilized to determine the Antimagic Labeling in Interconnection networks where the routing servers acting as its vertices and the connection between them acts as edges to get the best possible solution for building a network with computational power and the ability to send and receive messages over cabled or wireless communication links.

Distance antimagic labeling of circulant graphs

A distance antimagic labeling of graph $ G = (V, E) $ of order $ n $ is a bijection $ f:V(G)\rightarrow \{1, 2, \ldots, n\} $ with the property that any two distinct vertices $ x $ and $ y $ satisfy $ \omega(x)\ne\omega(y) $, where $ \omega(x) $ denotes the open neighborhood sum $ \sum_{a\in N(x)}f(a) $ of a vertex $ x $. In 2013, Kamatchi and Arumugam conjectured that a graph admits a distance antimagic labeling if and only if it contains no two vertices with the same open neighborhood. A circulant graph $ C(n; S) $ is a Cayley graph with order $ n $ and generating set $ S $, whose adjacency matrix is circulant. This paper provides partial evidence for the conjecture above by presenting distance antimagic labeling for some circulant graphs. In particular, we completely characterized distance antimagic circulant graphs with one generator and distance antimagic circulant graphs $ C(n; \{1, k\}) $ with odd $ n $.

On the antimagicness of generalized edge corona graphs

Given a graph G, a function of assigning distinct labels {1,2,...,|E(G)|} to E(G) such that w(a)≠w(b), ∀ a,b∈V(G) is an antimagic labeling of G where w(a) indicates the vertex sum obtained by summing up all the labels assigned to the edges incident on the vertex a. Let G, Hi, 1≤i≤m be connected graphs such that E(G)={e1,e2,...,em}. A new graph is constructed from G, Hi, 1≤i≤m by adding all possible edges between the end vertices of ei and V(Hi), i∈{1,2,...,m}. The resulting graph is called the generalized edge corona of G and (H1,H2,...,Hm) which is denoted as G⋄(H1,H2,...,Hm). We prove G ⋄ (H1,H2,...,Hm) is antimagic under certain conditions using an algorithmic approach where G has only one vertex of maximum degree three (excluding spider graphs containing uneven legs) and |V(Hi)|≥2, i∈{1,2,...,m}.

H - V -Super-Strong- ( a , d ) -antimagic decomposition of complete bipartite graphs

An H -(a,d)-antimagic labeling in a H -decomposable graph G is a bijection f : V ( G ) ∪ E ( G ) → { 1 , 2 , … , p + q } such that ∑ f ( H 1 ) , ∑ f ( H 2 ) , ⋯ , ∑ f ( H h ) forms an arithmetic progression with difference d and first element a . f is said to be H - V -super- ( a , d ) -antimagic if f ( V ( G ) ) = { 1 , 2 , … , p } . Suppose that V ( G ) = U ( G ) ∪ W ( G ) with | U ( G ) | = m and | W ( G ) | = n . Then f is said to be H - V -super-strong- ( a , d ) -antimagic labeling if f ( U ( G ) ) = { 1 , 2 , … , m } and f ( W ( G ) ) = { m + 1 , m + 2 , … , ( m + n = p ) } . A graph that admits a H - V -super-strong- ( a , d ) -antimagic labeling is called a H - V -super-strong- ( a , d ) -antimagic decomposable graph. In this paper, we prove that complete bipartite graphs K m , n are H - V -super-strong- ( a , d ) -antimagic decomposable with both m and n are even.

On the study of rainbow antimagic connection number of comb product of tree and complete bipartite graph

Rainbow antimagic coloring is the combination of antimagic labeling and rainbow coloring. The smallest number of colors induced from all edge weights of antimagic labeling is the rainbow antimagic connection number of [Formula: see text], denoted by [Formula: see text]. Given a graph [Formula: see text] with vertex set [Formula: see text] and edge set [Formula: see text], the function [Formula: see text] from [Formula: see text] to [Formula: see text] is a bijective function. The associated weight of an edge [Formula: see text] under [Formula: see text] is [Formula: see text]. A path [Formula: see text] in the vertex-labeled graph [Formula: see text] is said to be a rainbow [Formula: see text] path if for any two edges [Formula: see text] it satisfies [Formula: see text]. The function [Formula: see text] is called a rainbow antimagic labeling of [Formula: see text] if there exists a rainbow [Formula: see text] path for every two vertices [Formula: see text]. When we assign each edge [Formula: see text] with the color of the edge weight [Formula: see text], we say the graph [Formula: see text] admits a rainbow antimagic coloring. In this paper, we show the new lower bound and exact value of the rainbow antimagic connection number of comb product of any tree and complete bipartite graph.

On local distance antimagic labeling of graphs

Let G = ( V , E ) be a graph of order n and let f : V → { 1 , 2 … , n } be a bijection. For every vertex v ∈ V , we define the weight of the vertex v as w ( v ) = ∑ x ∈ N ( v ) f ( x ) where N(v) is the open neighborhood of the vertex v. The bijection f is said to be a local distance antimagic labeling of G if w ( u ) ≠ w ( v ) for every pair of adjacent vertices u , v ∈ V . The local distance antimagic labeling f defines a proper vertex coloring of the graph G, where the vertex v is assigned the color w(v). We define the local distance antimagic chromatic number χ l d ( G ) to be the minimum number of colors taken over all colorings induced by local distance antimagic labelings of G. In this paper we obtain the local distance antimagic labelings for several families of graphs including the path Pn , the cycle Cn , the wheel graph Wn , friendship graph Fn , the corona product of graphs G ° K m ¯ , complete multipartite graph and some special types of the caterpillars. We also find upper bounds for the local distance antimagic chromatic number for these families of graphs.

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 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.