3,785 research outputs found

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

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    A graph GG is called edge-magic if there exists a bijective function f:V(G)E(G){1,2,,V(G)+E(G)}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(u)+f(v)+f(uv)f\left(u\right) + f\left(v\right) + f\left(uv\right) is a constant (called the valence of ff) for each uvE(G)uv\in E\left( G\right) . If f(V(G))={1,2,,V(G)}f\left(V \left(G\right)\right) =\left\{1, 2, \ldots , \left\vert V\left( G\right) \right\vert \right\}, then GG 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 μs(G) \mu_{s}\left(G\right) of a graph GG is defined to be either the smallest nonnegative integer nn with the property that GnK1G \cup nK_{1} is super edge-magic or ++ \infty if there exists no such integer nn. On the other hand, the edge-magic deficiency μ(G) \mu\left(G\right) of a graph GG is the smallest nonnegative integer nn for which GnK1G\cup nK_{1} is edge-magic, being μ(G) \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 K1,nK_{1,n}

    Recent studies on the super edge-magic deficiency of graphs

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    A graph GG is called edge-magic if there exists a bijective function f:V(G)E(G){1,2,,V(G)+E(G)}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(u)+f(v)+f(uv)f\left(u\right) + f\left(v\right) + f\left(uv\right) is a constant for each uvE(G)uv\in E\left( G\right) . Also, GG is said to be super edge-magic if f(V(G))={1,2,,V(G)}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 μs(G) \mu_{s}\left(G\right) of a graph GG is defined to be either the smallest nonnegative integer nn with the property that GnK1G \cup nK_{1} is super edge-magic or ++ \infty if there exists no such integer nn. In this paper, we introduce the parameter l(n)l\left(n\right) as the minimum size of a graph GG of order nn for which all graphs of order nn and size at least l(n)l\left(n\right) have μs(G)=+\mu_{s} \left( G \right)=+\infty , and provide lower and upper bounds for l(G)l\left(G\right). Imran, Baig, and Fe\u{n}ov\u{c}\'{i}kov\'{a} established that for integers nn with n0(mod4)n\equiv 0\pmod{4}, μs(Dn)3n/21 \mu_{s}\left(D_{n}\right) \leq 3n/2-1, where DnD_{n} is the cartesian product of the cycle CnC_{n} of order nn and the complete graph K2K_{2} of order 22. We improve this bound by showing that μs(Dn)n+1 \mu_{s}\left(D_{n}\right) \leq n+1 when n4n \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

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    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

    Super edge-magic deficiency of join-product graphs

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    A graph GG is called \textit{super edge-magic} if there exists a bijective function ff from V(G)E(G)V(G) \cup E(G) to {1,2,,V(G)E(G)}\{1, 2, \ldots, |V(G) \cup E(G)|\} such that f(V(G))={1,2,,V(G)}f(V(G)) = \{1, 2, \ldots, |V(G)|\} and f(x)+f(xy)+f(y)f(x) + f(xy) + f(y) is a constant kk for every edge xyxy of GG. Furthermore, the \textit{super edge-magic deficiency} of a graph GG is either the minimum nonnegative integer nn such that GnK1G \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

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    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 xyE(G)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

    Problemas abiertos sobre etiquetamientos super edge-magic y temas relacionados

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    El tema de los etiquetamientos de grafos ha experimentado un fuerte impulso en los últimos 40 años, muestra de ello son los dos libros dedicados en exclusiva a ellos, un completísimo artículo ”survey” y más de 1000 artículos en la literatura. En este artículo exploramos algunas preguntas abiertas sobre etiquetamientos super edge-magic. Nos interesa particularmente este tipo de etiquetamientos, debido a la cantidad de relaciones que poseen con otras clases de etiquetamientos, principalmente los graciosos y los armónicos.Preprin

    Magic and antimagic labeling of graphs

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    "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

    PENGEMBANGAN BAHAN AJAR MATA KULIAH STUKTUR DATA BERBASIS WEB

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    Internet merupakan sebuah revolusi dalam perkembangan teknologi digital yang ditandai dengan terjadinya konvergensi antara teknologi komunikasi, komputer, dan penyiaran (broadcasting) menjadi sebuah teknologi informasi. Internet menjadi jaringan informasi dan komunikasi global pada masa kini
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