Perfect (super) Edge-Magic Crowns

A graph G is called edge-magic if there is a bijective function f from the set of vertices and edges to the set {1,2,…,|V(G)|+|E(G)|} such that the sum f(x)+f(xy)+f(y) for any xy in E(G) is constant. Such a function is called an edge-magic labelling of G and the constant is called the valence. An edge-magic labelling with the extra property that f(V(G))={1,2,…,|V(G)|} is called super edge-magic. A graph is called perfect (super) edge-magic if all theoretical (super) edge-magic valences are possible. In this paper we continue the study of the valences for (super) edge-magic labelings of crowns Cm¿K¯¯¯¯¯n and we prove that the crowns are perfect (super) edge-magic when m=pq where p and q are different odd primes. We also provide a lower bound for the number of different valences of Cm¿K¯¯¯¯¯n, in terms of the prime factors of m.Postprint (updated version

A SURVEY OF DISTANCE MAGIC GRAPHS

In this report, we survey results on distance magic graphs and some closely related graphs. A distance magic labeling of a graph G with magic constant k is a bijection l from the vertex set to {1, 2, . . . , n}, such that for every vertex x Σ l(y) = k,y∈NG(x) where NG(x) is the set of vertices of G adjacent to x. If the graph G has a distance magic labeling we say that G is a distance magic graph. In Chapter 1, we explore the background of distance magic graphs by introducing examples of magic squares, magic graphs, and distance magic graphs. In Chapter 2, we begin by examining some basic results on distance magic graphs. We next look at results on different graph structures including regular graphs, multipartite graphs, graph products, join graphs, and splitting graphs. We conclude with other perspectives on distance magic graphs including embedding theorems, the matrix representation of distance magic graphs, lifted magic rectangles, and distance magic constants. In Chapter 3, we study graph labelings that retain the same labels as distance magic labelings, but alter the definition in some other way. These labelings include balanced distance magic labelings, closed distance magic labelings, D-distance magic labelings, and distance antimagic labelings. In Chapter 4, we examine results on neighborhood magic labelings, group distance magic labelings, and group distance antimagic labelings. These graph labelings change the label set, but are otherwise similar to distance magic graphs. In Chapter 5, we examine some applications of distance magic and distance antimagic labeling to the fair scheduling of tournaments. In Chapter 6, we conclude with some open problems

Product of digraphs, (super) edge-magic valences and related problems

Discrete Mathematics, and in particular Graph Theory, has gained a lot of popularity during the last 7 decades. Among the many branches in Graph Theory, graph labelings has experimented a fast development, in particular during the last decade. One of the very important type of labelings are super edge-magic labelings introduced in 1998 by Enomoto et al. as a particular case of edge-magic labelings, introduced in 1970 by Kotzig and Rosa. An edge-magic labeling is a bijective mapping from the set of vertices and edges to [1, |V(G)|+|E(G)|], such that the sum of the labels of each edge and the incident vertices to it is constant. The constant is called the valence of the labeling. The edge-magic labeling is called super edge-magic if the smallest labels are assigned to the vertices. In this thesis, we consider three problems related to (super) edge-magic labelings and (di)graph products in which we use a family of super edge-magic digraphs as a second factor of the product. The digraph product we use, the h-product, was introduced by Figueroa-Centeno et al. in 2008. It is a generalization of the Kronecker product of digraphs. In Chapter 2, we study the super edge-magicness of graphs of equal order and size either by providing super edge-magic labelings of some elements in the family or proving that these labelings do not exist. The negative results are specially interesting since these kind of results are not common in the literature. Furthermore, the few results found in this direction usually meet one of the following reasons: too many vertices compared with the number of edges; too many edges compared with the number of vertices; or parity conditions. In our case, all previous reasons fail. In Chapter 3, we enlarge the family of perfect (super) edge-magic crowns. A crown is obtained from a cycle by adding the same number of pendant edges to each vertex of the cycle. Intuitively speaking, a (super) edge-magic graphs is perfect (super) edge-magic if all possible theoretical valences occur. The main result of the chapter is that the crowns defined by a cycle of length pq, where p and q are different odd primes, are perfect (super) edge-magic. We also provided lower bounds for the number of edge-magic valences of crowns. For graphs of equal order and size, the odd and the even labelling construction allows to obtain two edge-magic labelings from a particular super edge-magic labeling. The name refers to the parity of the vertex labels. In Chapter 4, we begin by providing some properties of odd and even labelling construction related to the (super) edge-magic labeling and also with respect to the digraph product. We also get a new application of the h-product by interchanging the role of the factors. This allows us to consider the classical conjecture of Godbold and Slater with respect to valences of cycles with a different point of view than the ones existing. Finally, we devote Chapter 5 to study the problem of edge-magic valences of crowns, in which even cycles appear, and to establish a relationship between super edge-magic graphs and graph decompositions. Some lower bounds on the number of (super) edge-magic valences are also established.La Matemàtica Discreta, i en particular la Teoria de Grafs, han guanyat molta popularitat durant les últimes set dècades. Entre les moltes branques de la Teoria de Grafs, els etiquetatges de grafs han experimentat un ràpid desenvolupament, especialment durant l'última dècada. Un dels tipus d'etiquetatges més importants són els etiquetatges super branca-màgics introduïts el 1998 per Enomoto et al. com un cas particular d'etiquetatges branca-màgics, introduïts el 1970 per Kotzig i Rosa. Un etiquetatge branca-màgic és una aplicació bijectiva del conjunt de vèrtexs i branques a [1, |V(G)|+|E(G)|], de manera que la suma de les etiquetes de cada branca i els vèrtexs incidents a ella és constant. La constant s'anomena valència de l'etiquetatge. L'etiquetatge branca-màgic s'anomena super branca-màgic si les etiquetes més petites s'assignen als vèrtexs. En aquesta tesi, considerem tres problemes relacionats amb etiquetatges (super) branca-màgic i productes de digrafs, en els que intervé una família de grafs super branca-màgic com a segon factor del producte. El producte de digrafs que usem, el producte h, va ser introduït per Figueroa-Centeno et al. el 2008. És una generalització del producte de Kronecker de digraphs. En el Capítol 2, estudiem el caràcter super branca-màgic de grafs d’ordre igual a mida, ja sigui proporcionant etiquetatges super branca-màgics d'alguns elements de la família o demostrant que aquests tipus d’etiquetatges no existeixen. Els resultats negatius són especialment interessants ja que aquest tipus de resultats no són comuns en la literatura. A més, els pocs resultats trobats en aquesta direcció solen encabir-se en una de les raons següents: massa vèrtexs en comparació amb el nombre de branques; massa branques en comparació amb el nombre de vèrtexs; o condicions de paritat. En el nostre cas, totes les raons anteriors fracassen. En el Capítol 3, ampliem la família de corones (super) branca-màgiques perfectes. Una corona és el graf que s’obté a partir d’un afegint el mateix nombre de branques a cada vèrtex del cicle. Intuïtivament parlant, un graf (super) branca màgic és (super) branca màgic si es donen totes les possibles valències teòriques. El resultat principal del capítol és que les corones definides per un cicle de longitud pq, on p i q són primers senars diferents, són (super) branca màgics perfectes. També proporcionem cotes inferiors per a la quantitat de valències màgiques de corones. Per a grafs d'igual ordre i mida, la construcció de l'etiquetatge senar i parell permet obtenir dos etiquetatges branca-màgics a partir d'un etiquetatge super branca-màgic. El nom fa referència a la paritat de les etiquetes de vèrtex. Al capítol 4, comencem proporcionant algunes propietats de la construcció de l'etiquetatge senar i parell relacionades amb l'etiquetatge (super) branca-màgic del que proven i també al producte h de dígrafs. També obtenim una nova aplicació del producte h intercanviant el paper dels factors. Això ens permet considerar la conjectura de Godbold i Slater respecte a les valències dels cicles des d’un punt de vista diferent a les existents. Finalment, dediquem el Capítol 5 a estudiar el problema de les valències branca-màgiques de les corones, en les que apareixen cicles parells, i a establir una relació entre els grafs super branca-màgic i les descomposicions de grafs. També s'estableixen alguns cotes inferiors del nombre de valències (super) branca-màgiques.Postprint (published version

An Example Usage of Graph Theory in Other Scientific Fields: On Graph Labeling, Possibilities and Role of Mind/Consciousness

This paper provides insights into some aspects of the possibilities and role of mind, consciousness, and their relation to mathematical logic with the application of problem solving in the fields of psychology and graph theory. This work aims to dispel certain long-held notions of a severe psychological disorder and a well-known graph labeling conjecture. The applications of graph labelings of various types for various kinds of graphs are being discussed. Certain results in graph labelings using computer software are presented with a direction to discover more applications

Distance magic-type and distance antimagic-type labelings of graphs

Generally speaking, a distance magic-type labeling of a graph G of order n is a bijection f from the vertex set of the graph to the first n natural numbers or to the elements of a group of order n, with the property that the weight of each vertex is the same. The weight of a vertex x is defined as the sum (or appropriate group operation) of all the labels of vertices adjacent to x. If instead we require that all weights differ, then we refer to the labeling as a distance antimagic-type labeling. This idea can be generalized for directed graphs; the weight will take into consideration the direction of the arcs. In this manuscript, we provide new results for d-handicap labeling, a distance antimagic-type labeling, and introduce a new distance magic-type labeling called orientable Gamma-distance magic labeling. A d-handicap distance antimagic labeling (or just d-handicap labeling for short) of a graph G=(V,E) of order n is a bijection f from V to {1,2,...,n} with induced weight function w(x_{i})=\underset{x_{j}\in N(x_{i})}{\sum}f(x_{j}) \] such that f(x_{i})=i and the sequence of weights w(x_{1}),w(x_{2}),...,w(x_{n}) forms an arithmetic sequence with constant difference d at least 1. If a graph G admits a d-handicap labeling, we say G is a d-handicap graph. A d-handicap incomplete tournament, H(n,k,d) is an incomplete tournament of n teams ranked with the first n natural numbers such that each team plays exactly k games and the strength of schedule of the ith ranked team is d more than the i+1st ranked team. That is, strength of schedule increases arithmetically with strength of team. Constructing an H(n,k,d) is equivalent to finding a d-handicap labeling of a k-regular graph of order n. In Chapter 2 we provide general constructions for every d at least 1 for large classes of both n and k, providing breadth and depth to the catalog of known H(n,k,d)\u27s. In Chapters 3 - 6, we introduce a new type of labeling called orientable Gamma-distance magic labeling. Let Gamma be an abelian group of order n. If for a graph G=(V,E) of order n there exists an orientation of G and a companion bijection f from V to Gamma with the property that there is an element mu in Gamma (called the magic constant) such that \[ w(x)=\sum_{y\in N_{G}^{+}(x)}\overrightarrow{f}(y)-\sum_{y\in N_{G}^{-}(x)}\overrightarrow{f}(y)=\mu for every x in V where w(x) is the weight of vertex x, we say that G is orientable Gamma-distance magic}. In addition to introducing the concept, we provide numerous results on orientable Z_n distance magic graphs, where Z_n is the cyclic group of order n. In Chapter 7, we summarize the results of this dissertation and provide suggestions for future work