On super (a, 1)-edge-antimagic total labelings of regular graphs

A labeling of a graph is a mapping that carries some set of graph elements into numbers (usually positive integers). An (a,d)-edge-antimagic total labeling of a graph with p vertices and q edges is a one-to-one mapping that takes the vertices and edges onto the integers 1,2…,p+q, so that the sum of the labels on the edges and the labels of their end vertices forms an arithmetic progression starting at a and having difference d. Such a labeling is called super if the p smallest possible labels appear at the vertices. In this paper we prove that every even regular graph and every odd regular graph with a 1-factor are super (a,1)-edge-antimagic total. We also introduce some constructions of non-regular super (a,1)-edge-antimagic total graphs

Totally Magic Graphs

A total labeling of a graph with v vertices and e edges is defined as a one-to-one map taking the vertices and edges onto the integers 1, 2, · · · , v+e. Such a labeling is vertex magic if the sum of the label on a vertex and the labels on its incident edges is a constant independent of the choice of vertex, and edge magic if the sum of an edge label and the labels of the endpoints of the edge is constant. In this paper we examine graphs possessing a labeling that is simultaneously vertex magic and edge magic. Such graphs appear to be rare

Super (a,d)-edge-antimagic total labeling of connected Disc Brake graph

Super edge-antimagic total labeling of a graph $G=(V,E)$ with order $p$ and size $q$, is a vertex labeling $\{1,2,3,...p\}$ and an edge labeling $\{p+1,p+2,...p+q\}$ such that the edge-weights, $w(uv)=f(u)+f(v)+f(uv), uv \in E(G)$ form an arithmetic sequence and for a>0 and $d\geq 0$, where $f(u)$ is a label of vertex $u$, $f(v)$ is a label of vertex $v$ and $f(uv)$ is a label of edge $uv$. In this paper we discuss about super edge-antimagic total labelings properties of connective Disc Brake graph, denoted by $Db_{n,p}$. The result shows that a connected Disc Brake graph admit a super $(a,d)$-edge antimagic total labeling for $d={0,1,2}$, $n\geq 3$, n is odd and $p\geq 2$. It can be concluded that the result has covered all the feasible $d$

Pseudo-magic graphs

AbstractWe characterize graphs for which there is a labeling of the edges by pairwise different integer labels such that the sum of the labels of the edges incident with a vertex is independent of the particular vertex. We generalize to mixed graphs, and to labelings with values in an integral domain

SUPER (a,d)-EDGE ANTIMAGIC TOTAL LABELING OF PENTAGONAL CHAIN GRAPH

Abstract. A G graph of order p and size q is called an (a,d)-edge antimagic total if there exist a bijection f: V(G)E(G) {1,2,…,p+q} such that the edge-weights, w(uv)=f(u)+f(v)+f(uv), uv E(G), form an arithmetic sequence with first term a and common difference d. Such a graph G is called super if the smallest possible labels appear on the vertices. In this paper we study super (a, d)-edge-antimagic total properties of connected PCn by using deductive axiomatic and the pattern recognition method. The result shows that a connected pentagonal chain graphs admit a super (a,d)-edge antimagic total  labeling for d = 0,1,2 for n It can be concluded that the result of this research has covered all the feasible d. Key Words: (a,d)-edge antimagic vertex labeling, super (a,d)-edge antimagic total labeling, Pentagonal Chain Graph

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