Super edge-magic deficiency of join-product graphs

A graph $G$ is called \textit{super edge-magic} if there exists a bijective function $f$ from $V(G) \cup E(G)$ to $\{1, 2, \ldots, |V(G) \cup E(G)|\}$ such that $f(V(G)) = \{1, 2, \ldots, |V(G)|\}$ and $f(x) + f(xy) + f(y)$ is a constant $k$ for every edge $xy$ of $G$. Furthermore, the \textit{super edge-magic deficiency} of a graph $G$ is either the minimum nonnegative integer $n$ such that $G \cup nK_1$ is super edge-magic or $+\infty$ if there exists no such integer. \emph{Join product} of two graphs is their graph union with additional edges that connect all vertices of the first graph to each vertex of the second graph. In this paper, we study the super edge-magic deficiencies of a wheel minus an edge and join products of a path, a star, and a cycle, respectively, with isolated vertices.Comment: 11 page

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

Some results concerning the valences of (super) edge-magic graphs

A graph $G$ is called edge-magic if there exists a bijective function $f:V\left(G\right) \cup E\left(G\right)\rightarrow \left\{1, 2, \ldots , \left\vert V\left( G\right) \right\vert +\left\vert E\left( G\right) \right\vert \right\}$ such that $f\left(u\right) + f\left(v\right) + f\left(uv\right)$ is a constant (called the valence of $f$) for each $uv\in E\left( G\right)$. If $f\left(V \left(G\right)\right) =\left\{1, 2, \ldots , \left\vert V\left( G\right) \right\vert \right\}$, then $G$ is called a super edge-magic graph. A stronger version of edge-magic and super edge-magic graphs appeared when the concepts of perfect edge-magic and perfect super edge-magic graphs were introduced. The super edge-magic deficiency $\mu_{s}\left(G\right)$ of a graph $G$ is defined to be either the smallest nonnegative integer $n$ with the property that $G \cup nK_{1}$ is super edge-magic or $+ \infty$ if there exists no such integer $n$. On the other hand, the edge-magic deficiency $\mu\left(G\right)$ of a graph $G$ is the smallest nonnegative integer $n$ for which $G\cup nK_{1}$ is edge-magic, being $\mu\left(G\right)$ always finite. In this paper, the concepts of (super) edge-magic deficiency are generalized using the concepts of perfect (super) edge-magic graphs. This naturally leads to the study of the valences of edge-magic and super edge-magic labelings. We present some general results in this direction and study the perfect (super) edge-magic deficiency of the star $K_{1,n}$

Recent studies on the super edge-magic deficiency of graphs

A graph $G$ is called edge-magic if there exists a bijective function $f:V\left(G\right) \cup E\left(G\right)\rightarrow \left\{1, 2, \ldots , \left\vert V\left( G\right) \right\vert +\left\vert E\left( G\right) \right\vert \right\}$ such that $f\left(u\right) + f\left(v\right) + f\left(uv\right)$ is a constant for each $uv\in E\left( G\right)$. Also, $G$ is said to be super edge-magic if $f\left(V \left(G\right)\right) =\left\{1, 2, \ldots , \left\vert V\left( G\right) \right\vert \right\}$. Furthermore, the super edge-magic deficiency $\mu_{s}\left(G\right)$ of a graph $G$ is defined to be either the smallest nonnegative integer $n$ with the property that $G \cup nK_{1}$ is super edge-magic or $+ \infty$ if there exists no such integer $n$. In this paper, we introduce the parameter $l\left(n\right)$ as the minimum size of a graph $G$ of order $n$ for which all graphs of order $n$ and size at least $l\left(n\right)$ have $\mu_{s} \left( G \right)=+\infty$, and provide lower and upper bounds for $l\left(G\right)$. Imran, Baig, and Fe\u{n}ov\u{c}\'{i}kov\'{a} established that for integers $n$ with $n\equiv 0\pmod{4}$, $\mu_{s}\left(D_{n}\right) \leq 3n/2-1$, where $D_{n}$ is the cartesian product of the cycle $C_{n}$ of order $n$ and the complete graph $K_{2}$ of order $2$. We improve this bound by showing that $\mu_{s}\left(D_{n}\right) \leq n+1$ when $n \geq 4$ is even. Enomoto, Llad\'{o}, Nakamigawa, and Ringel posed the conjecture that every nontrivial tree is super edge-magic. We propose a new approach to attak this conjecture. This approach may also help to resolve another labeling conjecture on trees by Graham and Sloane

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

Magic and antimagic labeling of graphs

"A bijection mapping that assigns natural numbers to vertices and/or edges of a graph is called a labeling. In this thesis, we consider graph labelings that have weights associated with each edge and/or vertex. If all the vertex weights (respectively, edge weights) have the same value then the labeling is called magic. If the weight is different for every vertex (respectively, every edge) then we called the labeling antimagic. In this thesis we introduce some variations of magic and antimagic labelings and discuss their properties and provide corresponding labeling schemes. There are two main parts in this thesis. One main part is on vertex labeling and the other main part is on edge labeling."Doctor of Philosoph

Four Lessons in Versatility or How Query Languages Adapt to the Web

Exposing not only human-centered information, but machine-processable data on the Web is one of the commonalities of recent Web trends. It has enabled a new kind of applications and businesses where the data is used in ways not foreseen by the data providers. Yet this exposition has fractured the Web into islands of data, each in different Web formats: Some providers choose XML, others RDF, again others JSON or OWL, for their data, even in similar domains. This fracturing stifles innovation as application builders have to cope not only with one Web stack (e.g., XML technology) but with several ones, each of considerable complexity. With Xcerpt we have developed a rule- and pattern based query language that aims to give shield application builders from much of this complexity: In a single query language XML and RDF data can be accessed, processed, combined, and re-published. Though the need for combined access to XML and RDF data has been recognized in previous work (including the W3C’s GRDDL), our approach differs in four main aspects: (1) We provide a single language (rather than two separate or embedded languages), thus minimizing the conceptual overhead of dealing with disparate data formats. (2) Both the declarative (logic-based) and the operational semantics are unified in that they apply for querying XML and RDF in the same way. (3) We show that the resulting query language can be implemented reusing traditional database technology, if desirable. Nevertheless, we also give a unified evaluation approach based on interval labelings of graphs that is at least as fast as existing approaches for tree-shaped XML data, yet provides linear time and space querying also for many RDF graphs. We believe that Web query languages are the right tool for declarative data access in Web applications and that Xcerpt is a significant step towards a more convenient, yet highly efficient data access in a “Web of Data”