Resumen de Product of digraphs, (super) edge-magic valences and related problems
Prabu Mohan
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.
