Local Antimagic Labeling Research Articles

Graceful Local Antimagic Labeling of Graphs: A Pattern Analysis Using Python

Graph labeling is the process of assigning labels to vertices and edges under certain conditions. This paper investigates the graceful local antimagic labeling of various graph families, excluding symmetric labelings, using computational experiments and Python-based algorithms. Through these experiments, we identify new results and patterns within specific graph classes. The study expands on the existing literature by offering computational evidence, proposing algorithms for the verification of labelings, and exploring the relationship between the local antimagic labeling and the chromatic number. Our results increase the understanding of graph labeling and offer insights into its computational aspects.

On Bridge Graphs with Local Antimagic Chromatic Number 3

Let G=(V,E) be a connected graph. A bijection f:E→{1,…,|E|} is called a local antimagic labeling if, for any two adjacent vertices x and y, f+(x)≠f+(y), where f+(x)=∑e∈E(x)f(e), and E(x) is the set of edges incident to x. Thus, a local antimagic labeling induces a proper vertex coloring of G, where the vertex x is assigned the color f+(x). 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 present some families of bridge graphs with χla(G)=3 and give several ways to construct bridge graphs with χla(G)=3.

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.

Https://rsmams.org/journals/articleinfo.php?articleid=994&tag=seajmams

Consider a simple graph $G$ without $K_2$ component with vertex set $V$ and edge set $E$. A local antimagic labeling $f$ of $G$ is a one-to-one mapping of edges to distinct positive integers $1,2, \dots, |E|$ such that the weights of adjacent vertices are distinct, where weight of a vertex is sum of labels assigned to the edges incident to it. These weights of the vertex induced by local antimagic labeling result in a proper vertex coloring of the graph $G$. The local antimagic chromatic number of $G$, denoted as $\chi_{la}(G)$, as the smallest number of distinct weights obtained across all possible local antimagic labelings of $G$. In this paper, we explore the local antimagic chromatic numbers of various classes of graphs, including the union of certain graph families, the corona product of graphs, and the necklace graph. In addition, we provide constructions for infinitely many graphs for which $\chi_{la}(G)$ equals the chromatic number $\chi(G)$ of the graph.

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 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 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.

Local antimagic chromatic number for the corona product of wheel and null graphs

Let $G=(V,E)$ be a graph of order $p$ and size $q$ having no isolated vertices. A bijection $f\colon E\hm{\rightarrow}\left\{1,2,3,\ldots,q \right\}$ is called a local antimagic labeling if for all $uv\in E$, we have $w(u)\neq w(v)$, the weight $w(u)=\sum_{e\in E(u)}f(e)$, where $E(u)$ is the set of edges incident to $u$. A graph $G$ is local antimagic, if $G$ has a local antimagic labeling. The local antimagic chromatic number $\chi_{la}(G)$ is defined to be the minimum number of colors taken over all colorings of $G$ induced by local antimagic labelings of $G$. In this paper, we completely determine the local antimagic chromatic number for the corona product of wheel and null graphs.

On Local Antimagic Vertex Coloring for Complete Full t-ary Trees

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 labeling induces a proper vertex coloring of G where the vertex v is assigned the color (vertex sum) 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. It was conjectured [6] that for every tree T the local antimagic chromatic number l + 1 ≤ χla( T) ≤ l + 2, where l is the number of leaves of T. In this article we verify the above conjecture for complete full t-ary trees, for t ≥ 2. A complete full t-ary tree is a rooted tree in which all nodes have exactly t children except leaves and every leaf is of the same depth. In particular we obtain that the exact value for the local antimagic chromatic number of all complete full t-ary trees is l + 1 for odd t.

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].

Local antimagic vertex coloring for generalized friendship graphs

Let G = (V, E) be a graph of order p and size q without isolated vertices. A bijection f: E → {1, 2, … , q} is called a local antimagic labeling if w(u) ≠ w(v) for all uv ∈ E, where the vertex weight w(u) = Ʃ e∈E(u) f(e) and E(u) is the set of edges incident to the vertex u ∈ V. The local antimagic chromatic number χ la (G) is defined to be the minimum number of colors(vertex weights) taken over all colorings of G induced by local antimagic labelings of G. In this paper, we study the local chromatic number for generalized friendship graph of complete graphs and cycles.

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.

The Local Antimagic Total Chromatic Number of Some Wheel-Related Graphs

Let G=(V,E) be a connected graph with |V|=n and |E|=m. A bijection f:V(G)∪E(G)→{1,2,⋯,n+m} is called local antimagic total labeling if, for any two adjacent vertices u and v, ωt(u)≠ωt(v), where ωt(u)=f(u)+∑e∈E(u)f(e), and E(u) is the set of edges incident to u. Thus, any local antimagic total labeling induces a proper coloring of G, where the vertex x in G is assigned the color ωt(x). The local antimagic total chromatic number, denoted by χlat(G), is the minimum number of colors taken over all colorings induced by local antimagic total labelings of G. In this paper, we present the local antimagic total chromatic numbers of some wheel-related graphs, such as the fan graph Fn, the bowknot graph Bn,n, the Dutch windmill graph D4n, the analogous Dutch graph AD4n and the flower graph Fn.

Local vertex antimagic chromatic number of some wheel related graphs

Let G = (V,E) be a graph of order p and size q having no isolated vertices. A bijection ƒ : E → {1, 2, 3, ..., q} is called a local antimagic labeling if for all uv ∈ E we have w(u) ≠ w(v), the weight w(u) = ∑e∈E(u) f(e) where E(u) is the set of edges incident to u. A graph G is local antimagic if G has a local antimagic labeling. The local antimagic chromatic number χla(G) is defined to be the minimum number of colors taken over all colorings of G induced by local antimagic labelings of G. In this paper, we determine the local antimagic chromatic number for some wheel related graphs.

On $(a,d)$-edge local antimagic coloring number of graphs

For any graph $G=(V,E),$ the order and size of G are $p$ and $q$. A bijection $l$ from $V(G)$ to $\{1,2,..,p\}$ is called $(a,d)$-edge local antimagic labeling if for any two adjacent edges are not received the same edge-weight (color) and the set of all edge-weights are formed an arithmetic progression $\{a,a+d,a+2d,\dots,a+(c-1)d\}$, for some integers $a,d>0$ and $c$ is the number of distinct colors used in the proper coloring.} An edge-weight (color) $w(uv)$ is the sum of two end vertices labels, $w(uv)=f(u)+f(v),uv\in E(G).$ The $(a,d)$-edge local antimagic coloring number is the least color (edge-weight) used in any $(a,d)$-edge local antimagic labeling. In the present study, we introduce a new type of labeling and a parameter, also we obtain the $(a,d)$-edge local antimagic coloring number for {paths and wheel graph $W_n$,$n=3,4,5$. Moreover, we obtain an upper bound of the $(a,d)$-edge local antimagic coloring number for wheel $W_n,n\ge 6$.

The Local Antimagic Chromatic Numbers of Some Join Graphs

Let G=(V(G),E(G)) be a connected graph with n vertices and m edges. A bijection f:E(G)→{1,2,⋯,m} is an edge labeling of G. For any vertex x of G, we define ω(x)=∑e∈E(x)f(e) as the vertex label or weight of x, where E(x) is the set of edges incident to x, and f is called a local antimagic labeling of G, if ω(u)≠ω(v) for any two adjacent vertices u,v∈V(G). It is clear that any local antimagic labelling of G induces a proper vertex coloring of G by assigning the vertex label ω(x) to any vertex x of G. The local antimagic chromatic number of G, denoted by χla(G), is the minimum number of different vertex labels taken over all colorings induced by local antimagic labelings of G. In this paper, we present explicit local antimagic chromatic numbers of Fn∨K2¯ and Fn−v, where Fn is the friendship graph with n triangles and v is any vertex of Fn. Moreover, we explicitly construct an infinite class of connected graphs G such that χla(G)=χla(G∨K2¯), where G∨K2¯ is the join graph of G and the complement graph of complete graph K2. This fact leads to a counterexample to a theorem of Arumugam et al. in 2017, and our result also provides a partial solution to Problem 3.19 in Lau et al. in 2021.

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).

Local Antimagic Chromatic Number for Copies of Graphs

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 where the color of a vertex is its vertex-weight. This naturally leads to the concept of a local antimagic chromatic number. The local antimagic chromatic number is defined to be the minimum number of colors taken over all colorings of G induced by local antimagic labelings of G. In this paper, we estimate the bounds of the local antimagic chromatic number for disjoint union of multiple copies of a graph.

On local super antimagic total face coloring and the application in developing a cipher block chaining key

There is a specific complexity in the use of IOT, especially in maintaining a secure data transaction. We need a good cryptosystem, since the best encryption key relays on the management cryptosystem. The biggest problem, then, is how to encrypt the plaintext into a ciphertext as hard as possible. We attempt to use the local super antimagic total face coloring of graph in developing a cipher block chaining key. By a local super antimagic total face coloring, we mean a bijection from the set of vertices, edges, and faces to {1, 2, 3, … |V(G)|+|E(G)|+|F(G)|} such that any adjacent two faces f 1 and f 2 will receive a different weights w(f 1) ≠ w(f 2) for f 1, f 2 ∈ F(G). If we use the weights as the color of all faces, the local super antimagic total face labeling is said to be a local super antimagic total face coloring. The minimum number possible to color all faces is called the local antimagic total face chromatic number, denoted by γlatf (G). Once, this resulting coloring in hand, we can develop a cipher block chaining key. We describe the results in the following two algorithms.