Graceful Labeling Research Articles

Characterizations of kites as graceful graphs

We introduce and study an infinite family of graceful graphs, which we call kites. The kites are graphs where a path is joined with a graph "forming" a kite. We study and characterize three classes of the kites: kites formed by cycles known to be graceful, fan kites and lantern kites. Beside showing in a transparent way that all these graphs are graceful, we provide characterizations of these graphs among all simple graphs via three tools: via Sheppard's labelling sequences introduced in the 1970s and via labelling relations and graph chessboards. The latter are relatively new tools for the study of graceful graphs introduced by Haviar and Iva\v ska in 2015. The labelling relations are closely related to Sheppard's labelling sequences while the graph chessboards provide a~nice visualization of the graceful labellings.

Arithmetic Sequential Graceful Labeling For Arrow Graphs

Let G be a simple, finite, connected, undirected, non-trivial graph with 𝑝 vertices and 𝑞 edges. 𝑉(𝐺) be the vertex set and 𝐸(𝐺) be the edge set of 𝐺. Let 𝑓: 𝑉(𝐺) → {𝑎, 𝑎 + 𝑑, 𝑎 + 2𝑑, 𝑎 + 3𝑑, … , 𝑎 + 2𝑞𝑑} where a ≥ 0 and 𝑑 ≥ 1 is an injective function. If for each edge 𝑢𝑣 ∈ 𝐸(𝐺) , 𝑓 ∗ : 𝐸(𝐺) → {𝑑, 2𝑑, 3𝑑, 4𝑑, … , 𝑞𝑑} defined by 𝑓 ∗ (𝑢𝑣) = |𝑓(𝑢)− 𝑓(𝑣)| is a bijective function then the function 𝑓 is called arithmetic sequential graceful labeling. The graph with arithmetic sequential graceful labeling is called arithmetic sequential graceful graph. In this paper, we prove that one side arrow graphs 𝐴𝑅𝜂 2 , 𝐴𝑅𝜂 3 , 𝐴𝑅𝜂 5 and double-sided arrow graphs 𝐷(𝐴𝑅𝜂 2 ),𝐷(𝐴𝑅𝜂 3 ) are arithmetic sequential graceful graph

Arithmetic Sequential Graceful Labeling of Complete Bipartite Graph with Pendant Edges

Arithmetic Sequential Graceful Labeling of Complete Bipartite Graph with Pendant Edges

Application of odd hexagonal graceful labeling in cryptography

Application of odd hexagonal graceful labeling in cryptography

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.

Some Algorithms of Graph Theory in Cryptology

The inventive use of concepts from Graph Theory plays a significant role in hiding the original Plain-text for resulting in a significantly safe data transfer. In this work, the tree traversal algorithms like Inorder, Preorder, Postorder, Kruskal’s algorithm for making minimal spanning tree and the modified graph labelling scheme of graceful labelling allowing repetition of exactly one vertex label for certain graphs, have been employed to create highly hidden Cipher-texts. Encryption and decryption algorithms for all these methods are being presented in this work.

Pelabelan Super Graceful pada Graf B dan Graf serta Implementasinya pada Matlab

Suppose G is a graph with vertices p dan edges q. Super graceful labeling is a one-to-one function maping on f:V(G) È E(G)  {1,2, … , p + q} such that f(uv) = ½f(u) – f(v)½ differs for each edge uv Î E(G). A graph G is called super graceful if it can be labeled according to the definition of super graceful labeling. The graph B(m,n,k) is a graph consisteing of a path graph of length k conneting the star graph K(1,m) and K(1,n) at the pendant ends. Meanwhile, the graph Pn(1,2,…,n) is a path graph of length by n combining each vertex on the path graph with edges i to i members (1,2,…,n). In this journal, it will be shown that graph B(m,n,k) and graph Pn(1,2,…,n) are super graceful graphs.

Using path coloring of graphs for communication in social networks

An L (p1, p2, p3, … , pm)- labeling of a graph G, has the vertices of G assigned with non-negative integers, such that the vertices at distance j should have at least pj as their label difference. If m = 3 and p1 = 3, p2 = 2, p3 = 1, it is called an L (3, 2, 1)-labeling which is widely studied in the literature. In this paper, we define an L (3, 2, 1)-path coloring of G as a labeling g : V (G) → Z+ such that between every pair of vertices there exists at least one path P where in the labeling restricted to this path is an L (3, 2, 1)-labeling. Among the labels assigned to any vertex of G under g, the maximum label is called the span of g. The L (3, 2, 1)-connection number of a graph G, denoted by k3c (G) is defined as the minimum value of span of g taken over all such labelings g. We call graphs with the special property that k3c (G) = |V (G) | as L (3, 2, 1)-path graceful. In this paper, we obtain k3c (G) of graphs that possess a Hamiltonian path and carry forward the discussion to certain classes of graphs which do not possess a Hamiltonian path, which is novel to this paper. Although different kinds of labeling are studied in the literature with different mathematical constraints imposed, the idea of showing the existence of a graph with a given number as its minimum labeling number has rarely been addressed. We show that given any positive integer, there always exists an L (3, 2, 1)-path graceful graph with the given integer as its k3c (G), thus addressing the inverse question. Finally exploiting the fact that there is no gap on the k3c (G) number line, we give an application of path colorings for secure communication on social networking sites. Efforts to deploy graph coloring in task scheduling, interference-free transmission, etc have been dealt by earlier researchers. In this paper, we deploy the L (3, 2, 1)-path coloring technique defined by us for secure communication in social networks, which has not been dealt with so far.

Enhanced Security in Public Key Cryptography: A Novel Approach Combining Gaussian Graceful Labeling and NTRU Public Key Cryptosystem

This research explores an encryption system that combines the Nth-degree Truncated Polynomial Ring Unit (NTRU) public key cryptosystem with Gaussian Graceful Labeling. This process assigns distinct labels to a graph's vertices, resulting in successive Gaussian integers. The NTRU method offers enhanced security and efficient key exchange. The communication encryption process depends on integers P, a, and b, with P being the largest prime number in the vertex labeling. The original receivers are the vertex labeling with the largest prime number coefficient, while all other receivers receive messages from the sender. A polynomial algebraic mixing system and a clustering principle based on the abecedarian probability proposition are used in NTRU encryption and decryption. The choice of relatively prime integers p and q in NTRU plays a role in the construction of polynomial rings used for encryption and decryption, with specific choices and properties designed to ensure scheme security.

Odd Graceful Labeling of W -Tree W T ( n , k ) and its Disjoint Union

Let G=(V(G),E(G)) be a graph with p vertices and q edges. A graph G of size q is said to be odd graceful if there exists an injection λ:V(G)→0,1,2,…,2q−1 such that assigning each edge xy the label or weight |λ(x)–λ(y)| results in the set of edge labels being 1,3,5,…,2q−1. This concept was introduced in 1991 by Gananajothi. In this paper, we examine the odd graceful labeling of the W-tree, denoted as WT(n,k).

Graceful game on some graph classes

A graceful labeling of a graph G with m edges consists in labeling the vertices of G with distinct integers from 0 to m such that each edge is uniquely identified by the absolute difference of the labels of its endpoints. In this work, we study the graceful labeling problem in the context of maker-breaker graph games. The Graceful Game was introduced by Tuza, in 2017, as a two-players game on a connected graph in which the players, Alice and Bob, take moves labeling the vertices with distinct integers from 0 to m. Players are constrained to use only legal labelings (moves), that is, after a move, all edge labels are distinct. Alice’s goal is to obtain a graceful labeling for the graph, as Bob’s goal is to prevent it from happening. In this work, we study winning strategies for Alice and Bob in graph classes: paths, complete graphs, cycles, complete bipartite graphs, caterpillars, trees, gear graphs, web graphs, prisms, hypercubes, 2-powers of paths, wheels and fan graphs.

Heptagonal graceful labeling of caterpillar and path related graphs

Heptagonal graceful labeling of caterpillar and path related graphs

On k-super graceful graphs with extremal maximum vertex degree

Let (,)GVE be a simple and loopless graph with ||V vertices and ||E edges. For 1,k≥ a graph G is called k-super graceful if there is a bijective labeling :fVE∪→{|ikik≤≤+||V+||E−1} with ()fvu=|()()|fvfu− for every edge .vuE∈ In this paper, we study the existence of k-super graceful labeling of certain graphs with extremal maximum vertex degree.

ABSOLUTE MEAN GRACEFUL LABELING IN THE CONTEXT OF m-SPLITTING AND DEGREE SPLITTING GRAPHS

A graph G with q edges is said to be absolute mean graceful if there is a one-to-one function f from V (G) to the set {0,±1,±2,±3,...,±q} such that when each edge xy is assigned the label |f(x)−f(y)| 2 , then the resulting edge labels are distinct. In this paper, the absolute mean graceful labeling of m-splitting and degree splitting graphs of some graphs are investigated.

Arithmetic Sequential Graceful Labeling of Star Related Graphs – II

Arithmetic Sequential Graceful Labeling of Star Related Graphs – II

A Noval Approach on Picture Fuzzy Graceful Labeling in Fluid flows

A Noval Approach on Picture Fuzzy Graceful Labeling in Fluid flows

Encoding And Decoding Process Using Prime Graceful Labeling

In this paper, an interesting application of Prime graceful labeling in the field of coding theory is examined. Here, a new algorithm for encoding and decoding blended with prime graceful labeling is constructed and an interpretation for the proposed algorithm is also presented.

Graceful labeling construction for some special tree graph using adjacency matrix

Graceful labeling construction for some special tree graph using adjacency matrix

Graceful and felicitous labeling of Stem-Lotus graph

A graph obtained from a shell graph $C(2n+3, 2n)$ where $n \geq 1$ by adding a vertex in between each pair of adjacent vertices on the cycle, adding an edge in apex and two more chords is called a Stem-Lotus graph. In this paper, I have proved that a Stem-Lotus graph is graceful and also felicitous.

Graceful Labeling of Prime Index Graph of Group Zp × Zpn

The prime index graph π(G) of a finite group G is a special type of undirected simple graph whose vertex set is set of subgroups of G, in which two distinct vertices are adjacent if one has prime index in the other. Let p and q be distinct primes. In this paper, we establish that prime index graph of a finite cyclic p-group Zpn, a finite abelian group Zpn × Zq and a finite abelian p-group Zp × Zpn always have graceful labeling without any condition on n using the concept of path graph or p-layer ladder graph of size n + 1.