A note on edge – Graceful labeling for Corona and flower graph

Let G be a simple graph, let f : V(G)&#8594{1,2,...,|V(G)|} be a bijective mapping. The weight of v &#8712 V(G) is the sum of labels of all vertices adjacent to v . We say that f is a distance magic labeling of G if the weight of every vertex is the same constant k and we say that f is a handicap magic labeling of G if the weight of every vertex v is l + f(v) for some constant l. Graphs that allow such labelings are called distance magic or handicap, respectively. Distance magic and handicap labelings of regular graphs are used for scheduling incomplete tournaments. While distance magic labelings correspond to so called equalized tournaments, handicap labelings can be used to schedule incomplete tournaments that are more challenging to stronger teams or players, hence they increase competition and yield attractive schemes in which every games counts. We summarize known results on distance magic and handicap labelings and construct a new infinite class of 4-regular handicap graphs.

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

Labelling problems are very useful in modelling, solving problems in other areas and enhance the study of algebraic structures. Nowadays, a more practical, accurate and efficient way of labeling a graph is using some mathematical methods like integer programming or constraint programming. This will improve the scope and quality of research. Here, considering the applications of Graceful and Harmonious labelings, we introduce a few new types of labeling, namely, gracefully harmonious labeling, gracefully felicitous labeling, gracefully elegant labeling, gracefully strongly c-elegant labeling and gracefully semi-harmonious labeling. In this paper, we prove that some complete multi-partite graphs is gracefully harmonious, gracefully felicitous and gracefully semi-harmonious. Keywords: Labeling; multi-partite graph; gracefully harmonious; gracefully felicitous; gracefully semi-harmonious. 2010 Mathematics Subject Classification: 05C78

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.

Let G be a simple graph of order n. Prime labeling is a bijective function f:V(G)→{1,2,…,n} such that gcd⁡(f(u),f(v))=1 for every pair of adjacent vertices u,v in G. A graph G that satisfies the definition of prime labeling is called a prime graph. The Dutch windmill graph D_r^n is a graph obtained by taking n copies of cycle graph C_r with a vertex in common. The double quadrilateral graph DQ is a graph constructed from two copies of C_4 and identifying one edge from each of them. The graph obtained by taking n copies of DQ and identifying one vertex of degree 3 from each of them as a common central vertex is called the double quadrilateral Dutch windmill graph DQ_n, for n≥1. Furthermore, graphs D_r^n and DQ_n becomes the base graph to construct two new graph classes, namely graph P_2 [D_r^n] and flower double quadrilateral graph FDQ_n. Both graph classes, constructed from the Dutch windmill graph, also contain even cycles. From previous research, it is known that graphs P_2 [D_4^n] and flower double quadrilateral graph FDQ_n have odd harmonious labeling. However, the determination of prime labeling on both classes is still an open problem. In this paper, we show that two classes of graphs constructed from Dutch windmill graphs with even cycles, namely graphs P_2 [D_4^n] and flower double quadrilateral graphs FDQ_n for n≥1 have a prime labeling. The result of this research shows that these graphs are prime graphs.

The Graceful Tree Conjecture stated by Rosa in the mid 1960s says that every tree can be gracefully labelled. It is one of the best known open problems in Graph Theory. The conjecture has caused a great interest in the study of gracefulness of simple graphs and has led to many new contributions to the list of graceful graphs. However, it has to be acknowledged that not much is known about the structure of graceful graphs after 55 years. Our paper adds an infinite family of classes of graceful graphs to the list of known simple graceful graphs. We introduce classes of \(k\)-enriched fan graphs \(kF_n\) for all integers \(k, n\ge 2\) and we prove that these graphs are graceful. Moreover, we provide characterizations of the \(k\)-enriched fan graphs \(kF_n\) among all simple graphs via Sheppard's labelling sequences introduced in the 1970s, as well as via labelling relations and graph chessboards. These last approaches are new tools for the study of graceful graphs introduced by Haviar and Ivaska in 2015. The labelling relations are closely related to Sheppard's labelling sequences while the graph chessboards provide a nice visualization of graceful labellings. We close our paper with an open problem concerning another infinite family of extended fan graphs.

<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>

Several of the most studied graph labelings are injective functions, this constraint precludes some graphs from admitting such labelings; a well-known example is given by the family of trees that cannot be harmoniously labeled. In order to study the existence of these labelings for certain graphs, the injectivity constraint is often dropped. In this work we eliminate this condition for two different, but related, additive vertex labelings such as the harmonious and arithmetic labelings. The new labelings are called semi harmonious and semi arithmetic. We consider some families of graphs that do not admit the injective versions of these labelings, among the graphs considered here we have cycles and other cycle-related graphs, including the analysis of some operations like the Cartesian product and the vertex or edge amalgamation; in addition, we prove that all trees admit a semi harmonious labeling. Something similar is done with the concept of arithmetic labeling, studying finite unions of semi arithmetic graphs together with some general results.

A graph H is a prime distance graph if there exists an one-to-one function h : V (H ) \(\to\) Z such that for any two adjacent vertices x and y, the integer |h(x) - h(y)| is a prime. So H is a prime distance graph if and only if there exists a prime distance labeling of H. If the edge labels of H are also distinct, then h becomes a distinct prime distance labeling of H and H is called a distinct prime distance graph. The generalized Petersen graphs P (n, k) are defined to be a graph on 2n (n \(\ge\) 3) vertices with V (P (n, k)) = {vi , ui : 0 \(\le\) i \(\le\)n - 1} and E(P (n, k)) = {vi vi+1 , vi ui , ui ui+k : 0 \(\le\) i \(\le\) n - 1, subscripts modulo n}. In this chapter, we give a concise overview on these two labeling, specifically, highlight that the generalized Petersen graphs P (n, 3) permit a prime distance labeling for all even n > 5 and conjecture that P (n, 2) and P (n, 3) admit a prime distance labeling for any n \(\ge\) 5 and all odd n \(\ge\) 5, respectively. Moreover, the cycle Cn admits a distinct prime distance labeling for all n \(\ge\) 3.

One important problem in graph theory is graph coloring or graph labeling.Labeling problem is a well-studied problem due to its wide applications, especially in frequency assignment in(mobile) communication systems, coding theory, x-ray crystallography, radar, circuit design, etc.For two non-negative integers, labeling of a graph is a function from the node set to the set of non-negative integers.For any two distinct vertices and of , a radio labeling is an injective function : 0 such that , | | 1 where is the diameter of .The radio number of , is the maximum value of taken overall radio labelings of .This paper determines the radio number of double triangular snake graphs.

Let G = (V (G), E(G)) be a graph with q edges. A function f is called harmonious labeling of graph G if f: V →¨{0, 1, 2,…, q-1} is injective and the induced function f*: E → {0, 1, 2,…, q} defined as f*(uv) = (f(u) + f(v))(mod q) is bijective. A graph which admits harmonious labeling is called harmonious graph. In this paper we prove that the jewel graph, triangular ladder graph, special flower graph, duplicating all the vertex of mK1, in P2+mK1, T(Pn)Kcm are harmonious graphs.

Let G be a graph. A bijection f:V→ {1,2,…..|V|} is called a prime labeling [3] if for each edge e=uv in E, we have GCD{ f(u),f(v)}=1. A graph that admits a prime labeling is said to be a prime graph. In this paper we show that bull graph admits Prime labeling in the context of variety graph operations namely duplication of vertex, fusion of vertices and Switching in Bull graph.

For any non-trivial abelian group A under addition a graph $G$ is said to be $A$-textit{magic} if there exists a labeling $f:E(G) rightarrow A-{0}$ such that, the vertex labeling $f^+$ defined as $f^+(v) = sum f(uv)$ taken over all edges $uv$ incident at $v$ is a constant. An $A$-textit{magic} graph $G$ is said to be $Z_k$-magic graph if the group $A$ is $Z_k$ the group of integers modulo $k$. These $Z_k$-magic graphs are referred to as $k$-textit{magic} graphs. In this paper we prove that the total graph, flower graph, generalized prism graph, closed helm graph, lotus inside a circle graph, $Godotoverline{K_m}$, $m$-splitting graph of a path and $m$-shadow graph of a path are $Z_k$-magic graphs.

This paper deals with the labeling of vertices and edges of a graph. Let G be a graph with vertex set V and edge set E, where |V| be the number of vertices and |E| edges of G. The two bijection methods for which we have designed algorithms are as follows. Initially A bijection λ 1:V∪E → {1, 2…|V| + |E|} is called a Vertex-Magic Total Labeling (VMTL) if there is a vertex magic constant vk such that the weight of vertex m is, λ 1(m) + Σ n εA (m) λ 1(mn) = vk, ∀ m εv Where A(m) is the set of vertices adjacent to x. In the similar fashion the bijection λ 2:V∪E → {1, 2…|V| + |E|} is called Edge-Magic Total Labeling (EMTL) if there is a edge magic constant ek such that the weight of an edge es(mn), λ 2(m) + λ 2(n) + λ 2 e(mn) = ek, ∀ e εE. A resultant Graph which consists of both VMTL and EMTL are said to be Total Magic Labeling (TML) for different vertex magic constant and edge magic constant values. Baker and Sawada proposed algorithms to find VMTLs on cycles and wheels. In this paper we enhanced these algorithms and also we proposed new algorithms to generate EMTLs and TMLs of cycles and wheels. We used the concept variations and sum set sequences to produce VMTLs and EMTLs on cycles and wheels. Also we designed modules to identify TML’s.KeywordsMagic labelingMagic labeling algorithmsMagic constantCyclesWheelsVertex Magic Total LabelingEdge Magic Total LabelingTotal Magic Labeling etc

Let $G=(V, E)$ be a graph and let $(A,+)$ be an Abelian group with identity elemento. Let $f: V \rightarrow A$ be a vertex labeling and $f^*: E \rightarrow A$ be the induced labeling of $f$, defined by $f^*\left(v_1 v_2\right)=f\left(v_1\right)+f\left(v_2\right)$ for all $v_1 v_2 \in E$. Then $f^*$ again induces a labeling say $f^{* *}: V \rightarrow A$ defined by $f^{* *}(v)=\sum_{v v_1 \in E} f^*\left(v v_1\right)$. A graph $G=(V, E)$ is said to be an Induced $A$-Magic Graph (IAMG) if there exists a non zero labeling $f: V \rightarrow A$ such that $f \equiv f^{* *}$. The function $f$, so obtained is called an Induced $A$-Magic Labeling (IAML) of $G$ and a graph which has no such Induced Magic Labeling is called a Non-induced magic graph. In this paper we discuss the existence of Induced Magic Labeling of some special graphs like $P_n, C_n, K_n$ and $K_{m, n}$.