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

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.

The graph G is a pair of sets consisting of a vertex set V(G) and an edge set E(G), denoted by G = (V (G),E(G)). Coloring a graph involves assigning colors to each vertex, edge, or region such that no adjacent vertices, edges, or regions share the same color. A bijective function f∶ V (G) → {1,2,3,...,|V (G)|} is called a local edge antimagic coloring if for any two adjacent edges e_1 and e_2, they have different weights, w(e_1) ≠ w(e_2), where e = uv ∈ E(G) and w(e) = f (u)+f (v). The chromatic number is the term used in the context of local antimagic coloring, referring to the minimum number of colors derived from local antimagic labeling. This research discusses the local antimagic edge coloring on the Gear Graph (G_n) and the Semi Parachute Graph (SP_(2n-1)). The aim of the research is to determine the chromatic number of local antimagic edge coloring χlea(G) for the researched graphs. The method used in this research is pattern detection to derive the general pattern. Based on the analysis, the chromatic number of local antimagic edge coloring is obtained for the Gear Graph (G_n) and the Semi Parachute Graph (SP_(2n-1)) are χlea (G_n)=n + 2 and χlea(SP_(2n-1) )=n+ 2.

The local antimagic labeling on a graph G with |V| vertices and |E| edges is defined to be an assignment f : E --> {1, 2,..., |E|} so that the weights of any two adjacent vertices u and v are distinct, that is, w(u)̸ ̸= w(v) where w(u) = Σe∈<sub>E(u)</sub> f(e) and E(u) is the set of edges incident to u. Therefore, any local antimagic labeling induces a proper vertex coloring of G where the vertex u is assigned the color w(u). The local antimagic chromatic number, denoted by χla(G), is the minimum number of colors taken over all colorings induced by local antimagic labelings of G. In this paper, we present the local antimagic chromatic number of unicyclic graphs that is the graphs containing exactly one cycle such as kite and cycle with two neighbour pendants.

Graceful graphs were first studied by Rosa in 1966. The Kotzig-Ringel graceful tree conjecture states that every tree has a graceful labeling. Aldred and McKay and others have used computer programs to show that trees of order up to 35 are graceful. Bagga et al. investigated algorithms for generating all graceful labelings of certain known classes of graceful graphs, including paths, cycles, and certain other classes of unicyclic graphs. The data generated by such algorithms has led to the discovery of new properties of such graceful labelings. In this paper we present a survey of graceful graph labeling algorithms and related complexity issues.DOI : http://dx.doi.org/10.22342/jims.0.0.14.1-9

Some problems on induced subgraphs

In this paper, we studied the Incident Binary Equivalent Decimal Edge Graceful Labeling (IBEDE) [1] and Strong Incident Binary Equivalent Decimal Edge Graceful Labeling (SIBEDE) [1] for Wheel Graph W1,n and Middle Graph of Wheel Graph M(W1,n). Also we obtained the Wheel Graph W1,n and its Middle Graph M(W1,n) for n ≥ 3 are IBEDE Graceful Labeling . And the Wheel Graph W1,n for n > 3and its Middle Graph M(W1,n) for n ≥ 3 are SIBEDE Graceful Labeling. Here the Ramsey number of IBEDE and SIBEDE Graceful Labeling for W1,5 is 3 and the Ramsey number of IBEDE and SIBEDE Graceful Labeling for M(W1,3) is 18.

<abstract><p>Graph labeling is a source of valuable mathematical models for an extensive range of applications in technologies (communication networks, cryptography, astronomy, data security, various coding theory problems). An edge $ \; \delta - $ graceful labeling of a graph $ G $ with $ p\; $ vertices and $ q\; $ edges, for any positive integer $ \; \delta $, is a bijective $ \; f\; $ from the set of edge $ \; E(G)\; $ to the set of positive integers $ \; \{ \delta, \; 2 \delta, \; 3 \delta, \; \cdots\; , \; q\delta\; \} $ such that all the vertex labels $ \; f^{\ast} [V(G)] $, given by: $ f^{\ast}(u) = (\sum\nolimits_{uv \in E(G)} f(uv)\; )\; mod\; (\delta \; k) $, where $ k = max (p, q) $, are pairwise distinct. In this paper, we show the existence of an edge $ \; \delta- $ graceful labeling, for any positive integer $ \; \delta $, for the following graphs: the splitting graphs of the cycle, fan, and crown, the shadow graphs of the path, cycle, and fan graph, the middle graphs and the total graphs of the path, cycle, and crown. Finally, we display the existence of an edge $ \; \delta- $ graceful labeling, for the twig and snail graphs.</p></abstract>

The paramount is ensuring future-proof and robust cybersecurity in the era of quantum computing. According to graph theory, the mathematical concept of Edge Antimagic Labelling (EATL) is leveraged by introducing a theoretical approach for encryption. Based on sum own label, a uniquely labelled edge is each graph edge. This work presents a quantum-resistant and secure encryption framework to exploit the combinational complexity, uniqueness, and inherent non-linearity of EATL. Construct the Quantum safe graphs as a special class of graphs, and the cryptographic key is generated to use the antimagic labelling that derives the dynamic edge weights. Both quantum and classical threats counter to improve the structural masking and key randomness by proposing improved security of EATL. Based on label prediction and graph isomorphism, a numerically hard problem space is provided via graph graph-theoretic foundation. The post-quantum, secure, and lightweight cryptographic models are designed with the powerful tool of EATL, established using graph theory.

A graph in this paper is nontrivial, finite, connected, simple, and undirected. Graph consists of a vertex set and edge set. Let u,v be two elements in vertex set, and q is the cardinality of edge set in G, a bijective function from the edge set to the first q natural number is called a vertex local antimagic edge labelling if for any two adjacent vertices and , the weight of is not equal with the weight of , where the weight of (denoted by ) is the sum of labels of edges that are incident to . Furthermore, any vertex local antimagic edge labelling induces a proper vertex colouring on where is the colour on the vertex . The vertex local antimagic chromatic number is the minimum number of colours taken over all colourings induced by vertex local antimagic edge labelling of . In this paper, we discuss about the vertex local antimagic chromatic number on disjoint union of some family graphs, namely path, cycle, star, and friendship, and also determine the lower bound of vertex local antimagic chromatic number of disjoint union graphs. The chromatic numbers of disjoint union graph in this paper attend the lower bound.

A graph G is an ordered pair of sets G(V, E), where V is a set of vertices and the elements of set E are usually called edges. The concept of graph labeling has recently gained a lot of popularity in the area of graph theory. his popularity is due not only to the mathematical challenges of graph labelings but also to the wide range of applications that graph labeling offer to other brances of science, for instance, x-ray, cryptography, coding theory, circuit design and communication network design. A graph labeling we mean an assignment of integers to elements of a graph such as vertex, edge, and both. Local antimagic of a graph was motivated by Arumugam et al.. Thus, we initiate to developed the concept of local antimagic vertex coloring for subdevided graphs. A graph G called sub-devided if the graph G be the graph obtained by inserting a vertex to each edge of the graph G. Definition the concept local antimagic vertex coloring is f : E(G) → {1, 2, 3…, |E(G)|} if for any two adjacent vertices a 1 and a 2, w(a 1) = w(a 2), where for v ∈ G, , where E(a) and V(a) are respectively the set of edges incident to a. The local antimagic vertex labeling induces a proper vertex coloring of graph G if each vertex a is assigned the color w(a). The minimum colors needed to coloring the vertices in graph G called local antimagic vertex chromatic number, denoted by χla (G). In this paper we study the local antimagic vertex coloring of sub-devided some special graph as follows SFn,m , SSn,m , SWn,m and SFn,m .

The structural information contentIg(X) of a graphX was treated in detail in three previous papers (Mowshowitz 1968a, 1968b, 1968c). Those investigations ofIg point up the desirability of defining and examining other entropy-like measures on graphs. To this end the chromatic information contentIc(X)of a graphX is defined as the minimum entropy over all finite probability schemes constructed from chromatic decompositions having rank equal to the chromatic number ofX. Graph-theoretic results concerning chromatic number are used to establish basic properties ofIc on arbitrary graphs. Moreover, the behavior ofIc on certain special classes of graphs is examined. The peculiar structural characteristics of a graph on which the respective behaviors of the entropy-like measuresIc andIg depend are also discussed.

Analyzing chromatic number in coloring problem is a tough topic in graph analysis. We focus on the basic theory for a particular type of chromatic number. This will give us insights on the basic topological structure guiding lots of networks in the coming trend of big data era. An adjacent vertex-distinguishing total k-coloring is a proper total k-coloring of a graph G such that for any two adjacent vertices, the set of colors appearing on the vertex and its incident edges are different. The smallest k for which such a coloring of G exists is called the adjacent vertex-distinguishing total chromatic number, and denoted by ?at(G). It has been proved that if the graph G satisfies ?(G)=3, then ?at(G)= 6. However, it is very difficult to determine whether ?at(G)= 5. In this paper, we focus on a special class of 3-regular graphs, the generalized Petersen graphs P(n, k), and show that ?at(P(n, k)) = 5, which improves the bound ?at (P(n, k))= 6.

We investigate the relation between the star-chromatic number χ(G) and the chromatic number χ(G) of a graphG. First we give a sufficient condition for graphs under which their starchromatic numbers are equal to their ordinary chromatic numbers. As a corollary we show that for any two positive integersk, g, there exists ak-chromatic graph of girth at leastg whose star-chromatic number is alsok. The special case of this corollary withg=4 answers a question of Abbott and Zhou. We also present an infinite family of triangle-free planar graphs whose star-chromatic number equals their chromatic number. We then study the star-chromatic number of An infinite family of graphs is constructed to show that for each ε>0 and eachm≥2 there is anm-connected (m+1)-critical graph with star chromatic number at mostm+ε. This answers another question asked by Abbott and Zhou.

A graceful labeling of a graph G with ‘q’ edges and vertex set V is an injection f: VG → {0,1,2,….q} with the property that the resulting edge labels are also distinct, where an edge incident with vertices u and v is assigned the label |fu – fv| . A graph which admits a graceful labeling is called a graceful graph. A Shell graph is defined as a cycle Cnwith n -3 chords sharing a common end point called the apex . Shell graphs are denoted as Cn, n- 3. A multiple shell is defined to be a collection of edge disjoint shells that have their apex in common. Hence a double shell consists of two edge disjoint shells with a common apex.

In a graph , define Pell labeling is a map : () {0,1, , -1} with an induced function * : () defined by * () = () + 2() for every () are all distinct where , 0. In addition to this a graph which admits Pell labeling concept is known as Pell graph.In this paper, the Pell labeling concepts applied for the following graphs such as splitting of Star, cycle with parallel chords, alternate double triangular Snake, ladder, Triangular ladder, diagonal ladder, shadow and splitting of path, bistar, subdivision of bistar, prism, , 2 and friendship graphs are studied.Our analysis contributes to the understanding of Pell labeling across a broad spectrum of graph configurations, highlighting its applicability and the unique characteristics of each considered graph family.