DE LA TESISSi un graf G admet un etiquetament super edge magic, aleshores G es diu que és un graf super edge màgic. La tesis està principalment enfocada a l'estudi del conjunt de grafs que admeten etiquetaments super edge magic així com també a desenvolupar relacions entre aquest tipus d'etiquetaments i altres etiquetaments molt estudiats com ara els etiquetaments graciosos i armònics, entre d'altres. De fet, els etiquetaments super edge magic serveixen com nexe d'unió entre diferents tipus d'etiquetaments, i per tant moltes relacions entre etiquetaments poden ser obtingudes d'aquesta forma. A la tesis també es proposa una nova manera de pensar en la ja famosa conjectura que afirma que tots els arbres admeten un etiquetament super edge magic. Això és, per a cada arbre T trobam un arbre super edge magic T' que conté a T com a subgraf, i l'ordre de T'no és massa gran quan el comparam amb l'ordre de T . Un problema de naturalesa similar al problema anterior, en el sentit que intentam trobar un graf super edge magic lo més petit possible i que contengui a cert tipus de grafs, i que ha estat completament resolt a la tesis es pot enunciar com segueix.Problema: Quin és un graf conexe G super edge magic d'ordre més petit que conté al graf complet Kn com a subgraf?.La solució d'aquest problema és prou interessant ja que relaciona els etiquetaments super edge magic amb un concepte clàssic de la teoria aditiva de nombres com són els conjunts de Sidon dèbils, també coneguts com well spread sets.De fet, aquesta no és la única vegada que el concepte de conjunt de Sidon apareix a la tesis. També quan a la tesis es tracta el tema de la deficiència , els conjunts de Sidon són d'una gran utilitat. La deficiencia super edge magic d'un graf és una manera de mesurar quan d'aprop està un graf de ser super edge magic. Tècnicament parlant, la deficiència super edge magic d'un graf G es defineix com el mínim número de vèrtexs aillats amb els que hem d'unirG perque el graf resultant sigui super edge magic. Si d'aquesta manera no aconseguim mai que el graf resultant sigui super edge magic, aleshores deim que la deficiència del graf és infinita. A la tesis, calculam la deficiència super edge magic de moltes families importants de grafs, i a més donam alguns resultats generals, sobre aquest concepte.Per acabar aquest document, simplement diré que al llarg de la tesis molts d'exemples que completen la tesis, i que fan la seva lectura més agradable i entenible han estat introduits. / OF THESISIf a graph G admits a super edge magic labeling, then G is called a super edge magic graph. The thesis is mainly devoted to study the set of graphs which admit super edge magic labelings as well as to stablish and study relations with other well known labelings.For instance, graceful and harmonic labelings, among others, since many relations among labelings can be obtained using super edge magic labelings as the link.In the thesis we also provide a new approach to the already famous conjecture that claims that every tree is super edge magic. We attack this problem by finding for any given tree T a super edge magic tree T' that contains T as a subgraph, and the order of T'is not too large if we compare it with the order of T .A similar problem to this one, in the sense of finding small host super edge magic graphs for certain type of graphs, which is completely solved in the thesis, is the following one.Problem: Find the smallest order of a connected super edge magic graph G that contains the complete graph Kn as a subgraph.The solution of this problem has particular interest since it relates super edge magic labelings with the additive number theoretical concept of weak Sidon set, also known as well spread set. In fact , this is not the only time that this concept appears in the thesis.Also when studying the super edge magic deficiency, additive number theory and in particular well spread sets have proven to be very useful. The super edge magic deficiency of graph is a way of measuring how close is graph to be super edge magic.Properly speaking, the super edge magic deficiency of a graph G is defined to be the minimum number of isolated vertices that we have to union G with, so that the resulting graph is super edge magic. If no matter how many isolated vertices we union G with, the resulting graph is never super edge magic, then the super edge magic deficiency is defined to be infinity. In the thesis, we compute the super edge magic deficiency of may important families of graphs and we also provide some general results, involving this concept.Finally, and in order to bring this document to its end, I will just mention that many examples that improve the clarity of the thesis and makes it easy to read, can be found along the hole work.

Aquest treball és una contribució a l'estudi de diferents problemes que sorgeixen de dues àrees fortament connexes de la Teoria de Grafs: etiquetaments i descomposicions. Molts etiquetaments de grafs deuen el seu origen als presentats l'any 1967 per Rosa. Un d'aquests etiquetaments, àmpliament conegut com a etiquetament graceful, va ser definit originalment com a eina per atacar la conjectura de Ringel, la qual diu que el graf complet d'ordre 2m+1 pot ser descompost en m copies d'un arbre donat de mida m. Aquí, estudiem etiquetaments relacionats que ens donen certes aproximacions a la conjectura de Ringel, així com també a una altra conjectura de Graham i Häggkvist que, en una forma dèbil, demana la descomposició d'un graf bipartit complet per un arbre donat de mida apropiada. Les principals contribucions que hem fet en aquest tema són la prova de la darrera conjectura per grafs bipartits complets del doble de mida essent descompostos per arbres de gran creixement i un nombre primer d'arestes, i la prova del fet que cada arbre és un subarbre gran de dos arbres pels quals les dues conjectures es compleixen respectivament. Aquests resultats estan principalment basats en una aplicació del mètode polinomial d'Alon. Un altre tipus d'etiquetaments, els etiquetaments magic, també són tractats aquí. Motivats per la noció de quadrats màgics de Teoria de Nombres, en aquest tipus d'etiquetaments volem asignar nombres enters a parts del graf (vèrtexs, arestes, o vèrtexs i arestes) de manera que la suma de les etiquetes assignades a certes subestructures del graf sigui constant. Desenvolupem tècniques basades en particions de certs conjunts d'enters amb algunes condicions additives per construir etiquetaments cycle-magic, un nou tipus d'etiquetament introduït en aquest treball i que estén la noció clàssica d'etiquetament magic. Els etiquetaments magic no donen cap descomposició de grafs, però les tècniques usades per obtenir-los estan al nucli d'un altre problema de descomposició, l'ascending subgraph decomposition (ASD). Alavi, Boals, Chartrand, Erdös i Oellerman, van conjecturar l'any 1987 que tot graf té un ASD. Aquí, estudiem l'ASD per grafs bipartits, una classe de grafs per la qual la conjectura encara no ha estat provada. Donem una condició necessària i una de suficient sobre la seqüència de graus d'un estable del graf bipartit de manera que admeti un ASD en que cada factor sigui un star forest. Les tècniques utilitzades estan basades en l'existència de branca-acoloriments en multigrafs bipartits. També tractem amb el sumset partition problem, motivat per la conjectura ASD, que demana una partició de [n] de manera que la suma dels elements de cada part sigui igual a un valor prescrit. Aquí donem la millor condició possible per la versió modular del problema que ens permet provar els millors resultats ja coneguts en el cas enter per n primer. La prova està de nou basada en el mètode polinomial. / This work is a contribution to the study of various problems that arise from two strongly connected areas of the Graph Theory: graph labelings and graph decompositions. Most graph labelings trace their origins to the ones presented in 1967 by Rosa. One of these labelings, widely known as the graceful labeling, originated as a means of attacking the conjecture of Ringel, which states that the complete graph of order 2m+1 can be decomposed into m copies of a given tree of size m. Here, we study related labelings that give some approaches to Ringel's conjecture, as well as to another conjecture by Graham and Häggkvist that, in a weak form, asks for the decomposition of a complete bipartite graph by a given tree of appropriate size. Our main contributions in this topic are the proof of the latter conjecture for double sized complete bipartite graphs being decomposed by trees with large growth and prime number of edges, and the proof of the fact that every tree is a large subtree of two trees for which both conjectures hold respectively. These results are mainly based on a novel application of the so-called polynomial method by Alon. Another kind of labelings, the magic labelings, are also treated. Motivated by the notion of magic squares in Number Theory, in these type of labelings we want to assign integers to the parts of a graph (vertices, edges, or vertices and edges) in such a way that the sums of the labels assigned to certain substructures of the graph remain constant. We develop techniques based on partitions of certain sets of integers with some additive conditions to construct cycle-magic labelings, a new brand introduced in this work that extends the classical magic labelings. Magic labelings do not provide any graph decomposition, but the techniques that we use to obtain them are the core of another decomposition problem, the ascending subgraph decomposition (ASD). In 1987, was conjectured by Alavi, Boals. Chartrand, Erdös and Oellerman that every graph has an ASD. Here, we study ASD of bipartite graphs, a class of graphs for which the conjecture has not been shown to hold. We give a necessary and a sufficient condition on the one sided degree sequence of a bipartite graph in order that it admits an ASD by star forests. Here the techniques are based on the existence of edge-colorings in bipartite multigraphs. Motivated by the ASD conjecture we also deal with the sumset partition problem, which asks for a partition of [n] in such a way that the sum of the elements of each part is equal to a prescribed value. We give a best possible condition for the modular version of the sumset partition problem that allows us to prove the best known results in the integer case for n a prime. The proof is again based on the polynomial method.

Let G be a directed graph with a total labeling. The additive arc-weight of an arc xy is the sum of the label on xy and the label on y. The additive directed vertex-weight of a vertex x is the sum of the label on x and the labels on all arcs with head at x. The graph is additive arc magic if all additive arc-weights are equal, and is additive directed vertex magic if all vertex-weights are equal. We provide a complete characterization of all graphs which permit an additive arc magic labeling. A complete characterization of all regular graphs which may be oriented to permit an additive directed vertex magic labeling is provided. The definition of the subtractive arc-weight of an arc xy is proposed, and a correspondence between graceful labelings and subtractive arc magic labelings is shown.

Let G be a directed graph with a total labeling. The additive arc-weight of an arc xy is the sum of the label on xy and the label on y. The additive directed vertex-weight of a vertex x is the sum of the label on x and the labels on all arcs with head at x. The graph is additive arc magic if all additive arc-weights are equal, and is additive directed vertex magic if all vertex-weights are equal. We provide a complete characterization of all graphs which permit an additive arc magic labeling. A complete characterization of all regular graphs which may be oriented to permit an additive directed vertex magic labeling is provided. The definition of the subtractive arc-weight of an arc xy is proposed, and a correspondence between graceful labelings and subtractive arc magic labelings is shown.