Super Edge-magic Labeling Research Articles

A Conjecture on Super Edge-Magic Total Labeling of 4-Cycle Books

A graph G is called cycle books B 4 , m , 2 if G consists of m cycles C 4 with a common path P 2 . Figueroa-Centeno, Ichishima, and Muntaner-Batle conjecture that the graph B 4 , m , 2 is super edge-magic total if and only if m is even or m ≡ 5 mod 8 . In this article, we prove this conjecture for m ≥ 36 and m = 0 mod (2).

Super edge-magic labeling for 𝒌-uniform, complete 𝒌-uniform and complete 𝒌-uniform 𝒌-partite hypergraphs

Super edge-magic labeling for 𝒌-uniform, complete 𝒌-uniform and complete 𝒌-uniform 𝒌-partite hypergraphs

On the (Consecutively) Super Edge-Magic Deficiency of Subdivision of Double Stars

LetGbe a finite, simple, and undirected graph with vertex setVGand edge setEG. A super edge-magic labeling ofGis a bijectionf:VG∪EG⟶1,2,…,VG+EGsuch thatfVG=1,2,…,VGandfu+fuv+fvis a constant for every edgeuv∈EG. The super edge-magic labelingfofGis called consecutively super edge-magic ifGis a bipartite graph with partite setsAandBsuch thatfA=1,2,…,AandfB=A+1,A+2,…,VG. A graph that admits (consecutively) super edge-magic labeling is called a (consecutively) super edge-magic graph. The super edge-magic deficiency ofG, denoted byμsG, is either the minimum nonnegative integernsuch thatG∪nK1is super edge-magic or+∞if there exists no suchn. The consecutively super edge-magic deficiency of a graphGis defined by a similar way. In this paper, we investigate the (consecutively) super edge-magic deficiency of subdivision of double stars. We show that, some of them have zero (consecutively) super edge-magic deficiency.

On the super edge-magic deficiency of some graphs

A graph G is called super edge-magic if there exists a bijection f:V(G)∪E(G)⟶{1,2,⋯,|V(G)|+|E(G)|}, where f(V(G))={1,2,⋯,|V(G)|}, such that f(u)+f(uv)+f(v) is a constant for every edge uv∈E(G). Such a case, f is called a super edge magic labeling of G. A bipartite graph G with partite sets A and B is called consecutively super edge-magic if there exists a super edge-magic labeling f with the property that f(A)={1,2,⋯,|A|} and f(B)={|A|+1,|A|+2,⋯,|V(G)|}. The super edge-magic deficiency of a graph G, denoted by μs(G), is either the minimum nonnegative integer n such that G∪nK1 is super edge-magic or +∞ if there exists no such n. The consecutively super edge-magic deficiency of a bipartite graph G, denoted by μc(G), is either the minimum nonnegative integer n such that G∪nK1 is consecutively super edge-magic or +∞ if there exists no such n. In this paper, we study the super edge-magic deficiency of some graphs. We investigate the (consecutively) super edge-magic deficiency of forests with two components. We also investigate the super edge-magic deficiency of a 2-regular graph 2C3∪Cn and join product of K1,n∪Pm with an isolated vertex.

Further results regarding super edge-magic graphs and solution to an open problem

Graph labeling admits a distinct place in Discrete Mathematics and Combinatorics. In 1998, Hikoe Enomoto et al. brought out the idea of super edge-magic (SEMC) labelling of graphs. In this article, we shall provide a complete solution to an open problem, regarding double stars St, t , proposed by W. D. Wallis & A. M. Marr [9], by exhibiting a super edge-magic labeling of St, t , for all possible values of t and t. We will also provide a SEMC labelling of union of cycles Cr being disjoint and various trees and exhibit super edge-magicness of some cases of Usmanian Ladders and planar graphs involving chains of C 5.

Edge Bimagic Labelings of Graphs K2, N, KN, 2, N, Bistar, nC4, and Double Wheel

Ngurahet. al. [2010] noted super magic coverings of fans and ladders. Ahmad et. al. [2011] enumerated super edge-magic labeling for the union of non-isomorphic graphs. Lopez et.al. [2012] suggested perfect super edge-magic graphs of path and cycle. Gao and Fan [2013] analyzed super edge-magic labelings of multi-star St(a1,…, an), Ali et. al. [2013] magic Labelings of type (a, b, c) of families of wheels.In this article, edge bimagiclabelings for bipartite complete graph, double bipartite complete graph, bistar merging with a path, one point union of finite copies of cycle having four vertices, and closed double wheel are found.

On Edge-magic Labelings of Forests

Given an n-vertex graph G = (V,E) with m edges, a labeling f of V ∪ E that uses all the labels in the set {1,2,...,n + m} is edge-magic if there is an integer k such that f(u) + f(v) + f(uv) = k for every edge uv ∈ E. Furthermore, if the labels in {1,2,...,n} are given to the vertices, then f is called super edge-magic. Kotzig [On magic valuations of trichromatic graphs, Reports of the CRM, 1971] started the investigation of super edge-magic labelings of forests. Following this line of research, we prove that some forests of stars admit a super edge-magic labeling and that some forests of caterpillars admit an edge-magic labeling.

On Edge Magic Total Labeling of (7, 3)-Cycle Books

A graph G is called (a, b)-cycle books B[(Ca, m), (Cb, n), Pt] if G consists of m cycles Ca and n cycles Cb with a common path Pt. In this article we show that the (7, 3)-cycle books B[(C7, 1), (C3, n), P2] admits edge-magic total labeling. In addition we prove that (a, 3)-cycle books B[(Ca, 1), (C3, n), P2] admits super edge-magic total labeling for a = 7 and extends the values of a to 4x − 1 for any positive integer x. Moreover we prove that the (7, 3)-cycle books B[(C7, 2), (C3, n), P2] admits super edge-magic total labeling.

Super edge-magic labeling of graphs: deficiency and maximality

A graph G of order p and size q is called super edge-magic if there exists a bijective function f from V(G) U E(G) to {1, 2, 3, ..., p+q} such that f(x) + f(xy) + f(y) is a constant for every edge $xy \in E(G)$ and f(V(G)) = {1, 2, 3, ..., p}. The super edge-magic deficiency of a graph G is either the smallest nonnegative integer n such that G U nK_1 is super edge-magic or +~ if there exists no such integer n. In this paper, we study the super edge-magic deficiency of join product graphs. We found a lower bound of the super edge-magic deficiency of join product of any connected graph with isolated vertices and a better upper bound of the super edge-magic deficiency of join product of super edge-magic graphs with isolated vertices. Also, we provide constructions of some maximal graphs, ie. super edge-magic graphs with maximal number of edges.

New Results on the (Super) Edge-Magic Deficiency of Chain Graphs

Let G be a graph of order v and size e. An edge-magic labeling of G is a bijection f:V(G)∪E(G)→{1,2,3,…,v+e} such that f(x)+f(xy)+f(y) is a constant for every edge xy∈E(G). An edge-magic labeling f of G with f(V(G))={1,2,3,…,v} is called a super edge-magic labeling. Furthermore, the edge-magic deficiency of a graph G, μ(G), is defined as the smallest nonnegative integer n such that G∪nK1 has an edge-magic labeling. Similarly, the super edge-magic deficiency of a graph G, μs(G), is either the smallest nonnegative integer n such that G∪nK1 has a super edge-magic labeling or +∞ if there exists no such integer n. In this paper, we investigate the (super) edge-magic deficiency of chain graphs. Referring to these, we propose some open problems.

On super edge-magic deficiency of certain Toeplitz graphs

A graph $G$ is called edge-magic if there exists a bijective function $\phi:V(G)\cup E(G)\to\{1, 2,\dots,|V(G)|+|E(G)|\}$ such that $\phi(x)+\phi(xy)+\phi(y)=c(\phi)$ is a constant for every edge $xy\in E(G)$, called the valence of $\phi$. Moreover, $G$ is said to be super edge-magic if $\phi(V(G))=\{1,2,\dots,|V(G)|\}.$ The super edge-magic deficiency of a graph $G$, denoted by $\mu_s(G)$, is the minimum nonnegative integer $n$ such that $G\cup nK_1,$ has a super edge-magic labeling, if such integer does not exist we define $\mu_s(G)$ to be $+\infty.$ In this paper, we study the super edge-magic deficiency of some Toeplitz graphs.

A Partial Solution to Linear Congruence Conjecture

Adams and Ponomarenko (Involv J Math 3(3):341–344, 2010) conjectured that when n is composite, \(k\ <\ n\) and \(gcd(a_1, a_2, . . . , a_k) \in {{\mathbb {Z}}_n^{\times }}\), then there exist distinct \(x_i \in {\mathbb Z_n}\) satisfying $$\begin{aligned} \sum _{i = 1}^{k} a_ix_i \equiv 1\ (\text{ in }\,\,{{\mathbb {Z}}_n}). \end{aligned}$$ In this paper, distinct solution has been constructed to the linear congruence when \(\sum _{i = 1}^{k} a_i = n-1\), using super edge-magic labeling of trees.

Super edge-magic labeling of [formula omitted]-node [formula omitted]-uniform hyperpaths and [formula omitted]-node [formula omitted]-uniform hypercycles

We generalize the notion of the super edge-magic labeling of graphs to the notion of the super edge-magic labeling of hypergraphs. For a hypergraph H with a finite vertex set V and a hyperedge set E, a bijective function f:V∪E→{1,2,3,…,|V|+|E|} is called a super edge-magic labeling if it satisfies (i) there exists a magic constant Λ such that f(e)+∑v∈ef(v)=Λ for all e∈E and (ii) f(V)={1,2,3,…,|V|}. A hypergraph admitting a super edge-magic labeling is said to be super edge-magic. In this paper, we show the equivalent form of this labeling, i.e, a hypergraph H is super edge-magic if and only if there exists a bijective function f:V→{1,2,3,…,|V|} such that {∑v∈ef(v)|e∈E} is the set of |E| consecutive integers. Finally, we define two classes of hypergraphs, namely m-nodek-uniform hyperpaths and m-nodek-uniform hypercycles which are denoted by mPn(k) and mCn(k), respectively. We show that under some conditions the hypergraphs mPn(k) and mCn(k) are super edge-magic.

On Super Edge-magic Total Labeling of Certain Classes of Graphs

A $(p, q)$- simple graph is edge-magic if there exists a bijective function $\lambda:V(G)\cup E(G)\rightarrow \{1, 2, \dots, p+q\}$ such that $\lambda(u)+\lambda(uv)+\lambda(v)=k$, for all edge $uv\in E(G),$ where $k$ is called the magic constant or sometimes the valence of $\lambda$. An edge-magic total labeling $\lambda$ is called super edge-magic total if $\lambda(V(G))=\{1, 2, \dots, p\}$. In this paper, we construct new classes of trees using w- trees and generalized combs and prove that they admit super edge magic total labeling. We also prove that the extended umbrella graphs admit super edge-magic total labeling.

The Jumping Knight and Other (Super) Edge-Magic Constructions

Let G be a graph of order p and size q with loops allowed. A bijective function \({f:V(G)\cup E(G)\rightarrow \{i\}_{i=1}^{p+q}}\) is an edge-magic labeling of G if the sum \({f(u)+f(uv)+f(v)=k}\) is independent of the choice of the edge uv. The constant k is called either the valence, the magic weight or the magic sum of the labeling f. If a graph admits an edge-magic labeling, then it is called an edge-magic graph. Furthermore, if the function f meets the extra condition that \({f(V(G))=\{i\}_{i=1}^{p}}\) then f is called a super edge-magic labeling and G is called a super edge-magic graph. A digraph D admits a labeling, namely l, if its underlying graph, und(D) admits l. In this paper, we introduce a new construction of super edge-magic labelings which are related to the classical jump of the knight on the chess game. We also use super edge-magic labelings of digraphs together with a generalization of the Kronecker product to get edge-magic labelings of some families of graphs.

Reverse super edge-magic strength of some new classes of graphs

In this paper, we introduce new concepts of reverse super edge-magic labeling and reverse super edge-magic strength of a graph G. A graph G is said to be reverse super edge-magic if there exists a bijection f : V ⋃ E → {1,2, …, p + q}, such that f (uv) – [f (u) + f (v)] is a constant, for all uv ∈ E and f (V) = {1,2,…, p}. Such a bijection is called a reverse super edge-magic labeling and the minimum of all constants is called a reverse super edge-magic strength of the graph G, where the minimum is taken over all reverse super edge-magic labelings of G. Also we obtain the reverse super edge-magic labelings and reverse super edge-magic strength of some well known graphs such as the y-tree Yn, the cycle C 2n+1, the generalized Petersen graph P(m, k) and the disconnected graph (2m + 1)C 3.

New problems related to the valences of (super) edge-magic labelings

A graph G of order p and size q is edge-magic if there is a bijective function f: V(G) ≼ E(G) → {i}p+qi=1 such that f(x) + f(xy) + f(y) = k, for all xy ∊ E(G). The function f is an edge-magic labeling of G and the sum k is called either the magic sum, the valence or the weight of f. Furthermore, if then f is a super edge-magic labeling of G. In this paper we study the valences that can be attained by (super) edge-magic labelings of some families of graphs.

On super edge-magic decomposable graphs

Let G be any graph and let {H i } i∈I be a family of graphs such that \(E\left( {H_i } \right) \cap E\left( {H_j } \right) = \not 0\) when i ≠ j, ∪ i∈I E(H i ) = E(G) and \(E\left( {H_i } \right) \ne \not 0\) for all i ∈ I. In this paper we introduce the concept of {H i } i∈I -super edge-magic decomposable graphs and {H i } i∈I -super edge-magic labelings. We say that G is {H i } i∈I -super edge-magic decomposable if there is a bijection β: V(G) → {1,2,..., |V(G)|} such that for each i ∈ I the subgraph H i meets the following two requirements: β(V(H i )) = {1,2,..., |V(H i )|} and {β(a) +β(b): ab ∈ E(H i )} is a set of consecutive integers. Such function β is called an {H i } i∈I -super edge-magic labeling of G. We characterize the set of cycles C n which are {H 1, H 2}-super edge-magic decomposable when both, H 1 and H 2 are isomorphic to (n/2)K 2. New lines of research are also suggested.

ENUMERATING SUPER EDGE-MAGIC LABELINGS FOR THE UNION OF NONISOMORPHIC GRAPHS

AbstractA super edge-magic labeling of a graph G=(V,E) of order p and size q is a bijection f:V ∪E→{i}p+qi=1 such that: (1) f(u)+f(uv)+f(v)=k for all uv∈E; and (2) f(V )={i}pi=1. Furthermore, when G is a linear forest, the super edge-magic labeling of G is called strong if it has the extra property that if uv∈E(G) , u′,v′ ∈V (G) and dG (u,u′ )=dG (v,v′ )&lt;+∞, then f(u)+f(v)=f(u′ )+f(v′ ). In this paper we introduce the concept of strong super edge-magic labeling of a graph G with respect to a linear forest F, and we study the super edge-magicness of an odd union of nonnecessarily isomorphic acyclic graphs. Furthermore, we find exponential lower bounds for the number of super edge-magic labelings of these unions. The case when G is not acyclic will be also considered.

Super Edge-magic Total Labeling in Extended Duplicate Graph of Path

In this paper, we prove that the Extended Duplicate Graph of a path is super edge-magic Keywords: Graph labeling, super edge-magic total labeling, Extended Duplicate Graphs