Graceful Labeling Research Articles

Graceful distance labeling for some particular graphs

Graceful distance labeling for some particular graphs

Odd graceful labeling on the Ilalang graph (Sn , 3) it’s variation

A graph called odd graceful graph if there is an injective mapping f:V(G) → {0,1,2,…,2q − 1} such that for each edge xy 6 E(G) labeled by |f(u) − f(v)|, and the results willform a label on the edge {1,3,5, …,2q − 1} which is bijective. Variabel q is the number of edges. This paper purpose is to construct the formula of odd graceful labelling on the ilalang graph (Sn , 3) and it’s variation.

FIBONACCI AND SUPER FIBONACCI GRACEFUL LABELLINGS OF SOME TYPES OF GRAPHS

We consider the basic theoretical information regarding the Fibonacci graceful graphs. An injective function is said a Fibonacci graceful labelling of a graph of a size , if it induces a bijective function on the set of edges , where by the rule , for any adjacent vertices A graph that allows such labelling is called Fibonacci graceful. In this paper, we introduce the concept of super Fibonacci graceful labelling, narrowing the set of vertex labels, i.e. Four types of problems to be studied are selected. In the problem of the first type, the following question is raised: is there a graph that allows a certain kind of labelling, and under what conditions does this take place? The problem of the second type is the problem of construction: it is necessary, for a given system of requirements for the graph, to construct (at least one) its labelling that would satisfy this system. The following two types of problems relate to enumeration problems: for a given graph, determine the number of different Fibonacci and / or super Fibonacci graceful labellings; build all the different labellings of a given kind. As a result of solving these problems, functions were found that generate Fibonacci and super Fibonacci graceful labellings for graphs of cyclic structure; necessary and sufficient conditions for the existence of Fibonacci graceful labelling for disjunctive union of cycles, super Fibonacci graceful labelling for cycles, Eulerian graphs are obtained; the number of non-equivalent labellings of the cycle is determined; conditions for the existence of a super Fibonacci graceful labelling of a one-point connection of arbitrary connected super Fibonacci graceful graphs … …, are presented

Odd-even graceful labeling of planar grid and prism graphs

A graph G with p vertices and q edges is said to be odd-even graceful graph, if there exist an injective function f : V(G) → {1, 3, 5, … , 2q + 1} such that induced function f * : E(G) → {2, 4, 6, … , 2q} defined by f *(uv) =| f (u) − f (v)| for every uv ∈ E(G) is a bijection. In this paper we proved that the planar grid and the prism admit odd-even graceful labeling.

On the Graceful Game

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, when each edge is assigned the absolute difference of the labels of its endpoints, all induced edge labels are distinct. Rosa established two well known conjectures: all trees are graceful (1966) and all triangular cacti are graceful (1988). In order to contribute to both conjectures we study these problems in the context of 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 turns labeling the vertices with distinct integers from 0 to m. Alice’s goal is to gracefully label the graph as Bob’s goal is to prevent it from happening. In this work, we present the first results in this area by showing winning strategies for Alice and Bob in complete graphs, paths, cycles, complete bipartite graphs, caterpillars, prisms, wheels, helms, webs, gear graphs, hypercubes and some powers of paths.

SUPER FIBONACCI GRACEFUL LABELING FOR GENERALIZED (a,m)-SHELL GRAPH AND MERGED WITH SOME GRAPHS

A graph $G$ with $p$ vertices and $q$ edges has super Fibonacci graceful labeling if there exists an injective map $f : V(G) \rightarrow \left\{F_{0}, F_{1},F_{2},\ldots F_{q}\right\}$ where $F_{k}$ is the $k^{th}$ Fibonacci number of the Fibonacci series such that its induced map $f^{+}: E(G) \rightarrow\left\{F_{1},F_{2},F_{3},\ldots F_{q}\right\}$ defined by $f^{+}(xy)$ =$\left|f(x) - f(y)\right|$ $\forall$ $xy \in G,$ is a bijective map. In this paper, we investigate the existence of super Fibonacci graceful labeling for the various types of $(a, m)$ - shell graph and $(a, m)$ - shell graph merged with some graphs.

EDGE ODD GRACEFUL LABELING OF SOME FLOWER PETAL GRAPHS

A labeling of a graph $G$ with $\alpha$ vertices and $\beta$ edges called an edge odd graceful labeling if there is an edge labeling with odd numbers to all edges such that each vertex is assigned a label which is the sum $\mod (2\gamma)$ of labels of edge incident on it, where $\gamma=max\{\alpha,\beta\}$ and the induced vertex labels are distinct. In this paper, we discussed about edge odd gracefulness of some special class of flower petal graphs.

Alpha graphs with different pendent paths

Graceful labelings are an effective tool to find cyclic decompositions of complete graphs and complete bipartite graphs. The strongest kind of graceful labeling, the α-labeling, is in the center of the research field of graph labelings, the existence of an α-labeling of a graph implies the existence of several, apparently non-related, other labelings for that graph. Furthermore, graphs with α-labelings can be combined to form new graphs that also admit this type of labeling. The standard way to combine these graphs is to identify every vertex of a base graph with a vertex of another graph. These methods have in common that all the graphs involved, except perhaps the base, have the same size. In this work, we do something different, we prove the existence of an α-labeling of a tree obtained by attaching paths of different lengths to the vertices of a base path, in such a way that the lengths of the pendent paths form an arithmetic sequence with difference one, where consecutive vertices of the base path are identified with paths which lengths are consecutive elements of the sequence. These α-trees are combined in several ways to generate new families of α-trees. We also prove that these trees can be used to create unicyclic graphs with an α-labeling. In addition, we show that the pendent paths can be substituted by equivalent α-trees to produce new α-trees, obtaining in this manner a quite robust category of α-trees.

A survey and a new class of graceful unicylic graphs

A graph G admits a graceful labeling if there is a one-to-one map f from the set of vertices of G to such that when an edge xy is assigned the label the resulting set of edge labels is When such a labeling exists, G is called graceful. Rosa showed that a cycle Cn () is graceful if and only if n is congruent to 0 or 3 modulo 4. Truszczyński conjectured that unicyclic graphs, except the cycles in Rosa’s result are graceful. Several classes of unicyclic graphs are known to be graceful. In this article, we present a survey of results related to Truszczyński’s conjecture. We also present a new class of graceful unicyclic graphs.

Radio even mean graceful labeling on some special graphs

Radio Even Mean Graceful Labeling of a connected graph \(G\) is a bijection \(\phi\) from the vertex set \(V(G)\) to \(\{2,4,6, \ldots 2|V|\}\) satisfying the condition \(d(s, t)+\left\lceil\frac{\phi(s)+\phi(t)}{2}\right\rceil \geq 1+\operatorname{diam}(G)\) for every \(\mathrm{s}, \mathrm{t} \in \mathrm{V}(\mathrm{G})\). A graph which admits radio even mean graceful labeling is called radio even mean graceful graph. In this paper we investigate the radio even mean graceful labeling on degree splitting of some special graphs.

On 0-Rotatable Graceful Caterpillars

An injection $$f :V(T) \rightarrow \{0,\ldots ,|E(T)|\}$$ of a tree T is a graceful labelling if $$\{|f(u)-f(v)| :uv \in E(T)\}=\{1,\ldots ,|E(T)|\}$$ . Tree T is 0-rotatable if, for any $$v \in V(T)$$ , there exists a graceful labelling f of T such that $$f(v)=0$$ . In this work, the following families of caterpillars are proved to be 0-rotatable: caterpillars with a perfect matching; caterpillars obtained by linking one leaf of the star $$K_{1,s-1}$$ to a leaf of a path $$P_n$$ with $$n \ge 3$$ and $$s \ge \lceil \frac{n}{2} \rceil $$ ; caterpillars with diameter five or six; and caterpillars T with $$\mathrm {diam}(T) \ge 7$$ such that, for every non-leaf vertex $$v \in V(T)$$ , the number of leaves adjacent to v is even and is at least $$2+2((\mathrm {diam}(T)-1)\bmod {2})$$ . These results reinforce the conjecture that all caterpillars with diameter at least five are 0-rotatable.

Approximation Algorithms for the Achromatic Number of Certain Graceful Graphs

Approximation Algorithms for the Achromatic Number of Certain Graceful Graphs

Dividing Graceful Labeling of Certain Tree Graphs

A tree is a connected acyclic graph on n vertices and m edges. graceful labeling of a tree defined as a simple undirected graph G(V,E) with order n and size m, if there exist an injective mapping that induces a bijective mapping defined by for each and . In this paper we introduce a new type of graceful labeling denoted dividing graceful then discuss this type of certain tree graphs .

Edge even graceful labeling of torus grid graph

We study the family of torus grid graphs. We also obtain necessary and sufficent conditions to be edge even graceful labeling for all of the cases of every member of this family.

Total difference chromatic numbers of graphs

Inspired by graceful labelings and total labelings of graphs, we introduce the idea of total difference labelings. A $k$-total labeling of a graph $G$ is an assignment of $k$ distinct labels to the edges and vertices of a graph so that adjacent vertices, incident edges, and an edge and its incident vertices receive different labels. A $k$-total difference labeling of a graph $G$ is a function $f$ from the set of edges and vertices of $G$ to the set $\{1,2,\ldots,k\}$, that is a $k$-total labeling of $G$ and for which $f(\{u,v\})=|f(u)-f(v)|$ for any two adjacent vertices $u$ and $v$ of $G$ with incident edge $\{u,v\}$. The least positive integer $k$ for which $G$ has a $k$-total difference labeling is its total difference chromatic number, $\chi_{td}(G)$. We determine the total difference chromatic number of paths, cycles, stars, wheels, gears and helms. We also provide bounds for total difference chromatic numbers of caterpillars, lobsters, and general trees.

G-GRACEFUL LABELING OF GRAPHS

Adv. Math. Sci. J. Vol. 9, No. 4 Open Access G-GRACEFUL LABELING OF GRAPHS A. Kumar, V. Kumar, K. Kumar, P. Gupta and Y. Khandelwal DOI: https://doi.org/10.37418/amsj.9.4.56 Full Text Abstract Abstract After a long period of scramble over analysis and investigations, the notion of graceful labeling came into existence. A mapping *** … page AMSJ9-N4-56 Read More »

Folding trees gracefully

When a graceful labeling of a bipartite graph assigns the smaller labels to the vertices of one of the stable sets of the graph, the assignment is called an α-labeling. Any graph that admits such a labeling is an α-graph. In this work we extend the concept of vertex amalgamation to generate a new class of α-graphs obtained by a sequence of k-vertex amalgamations of t copies of an α-tree. This procedure is also applied to any collection of α-trees such that any pair of trees in this collection have stable sets with the same cardinalities. We also use this idea on other types of α-graphs. In addition, we present a family of α-trees of even diameter formed with four caterpillars of the same size.

Odd-even gracefulness of splitting graph of some standard graphs

For a graph G the splitting graph S (G) is the graph obtained from G by adding a vertex v¢ for each vertex v of a graph G and join v¢ to all the vertices of G adjacent to v. In this paper, we show that the splitting graph of path Pn , star K 1, n , bi-star B(n), complete bipartite Kn, m and cycle Cn are odd-even graceful graphs.

Further results on edge even graceful labeling of the join of two graphs

In this paper, we investigated the edge even graceful labeling property of the join of two graphs. A function f is called an edge even graceful labeling of a graph G=(V(G),E(G)) with p=|V(G)| vertices and q=|E(G)| edges if f:E(G)→{2,4,...,2q} is bijective and the induced function f∗:V(G) →{0,2,4,⋯,2q−2 }, defined as f^{ast }(x) = ({sum nolimits }_{xy in E(G)} f(xy)~)~mbox{{mod}}~(2k) , where k=max(p,q), is an injective function. Sufficient conditions for the complete bipartite graph Km,n =mK1+nK1 to have an edge even graceful labeling are established. Also, we introduced an edge even graceful labeling of the join of the graph K1 with the star graph K1,n, the wheel graph Wn and the sunflower graph sfn for all n in mathbb {N}. Finally, we proved that the join of the graph overline {K}_{2}~ with the star graph K1,n, the wheel graph Wn and the cyclic graph Cn are edge even graceful graphs.

Hexagonal Snakes With Alpha Labeling Along A Path

The graceful labeling of a graph G with p vertices and q edges is an injection f: V (G) to {0, 1, 2… q} such that, when each edge uv is assigned the label |f (u)-f (v)|. The resulting edge set are {1, 2, 3 …q}. When graceful Labeling is applied to a graph it is known as a Gracful graph. In this paper, we introduce the graph HSn by arranging a series of Hexagons glued to each other by attaching the any one vertex of the first hexagon to the next one and the second hexagon to the third one and so on which is called a Hexagonal Snake. Thus, ‘n’ number of of hexagonal snakes are fixed along a path. We have applied graceful labeling to this Hexagonal Snake graph and have shown that it satisfies α-labeling.