Resumen de Difference vertex labelings
Christian Humberto Barrientos Diaz
The main part of this work is devoted to graceful graphs and graceful labelings.
We presented several families of graceful graphs. Some of the labelings exhibited satisfy the additional condition to be á-labelings. This kind of labeling can be used to obtain cyclic decompositions of K2kn+1 into isomorphic subgraphs of size n; they can also be used to decompose Knr,ns.
In addition, we can use them to produce odd graceful, 2-equitable and sequential labelings of graphs (which can be used to generate harmonious labelings).
Through this thesis, graphs with á-labelings are "combined" to produce larger graphs with suitable labelings.
Chapter 2 is devoted to the gracefulness of graphs obtained by the concatenation of certain types of blocks. Cyclic snakes are obtained by the concatenation of Cm. Some partial results related to their gracefulness can be found in the literature. We completely solved the case where the cycle has even size less than 14. Chain graphs appeared as a generalization of cyclic snakes. In this direction we prove that any chain graph whose blocks are complete bipartite graphs is graceful.
Chapter 3 is dedicated to graphs obtained by the corona operation. We prove that when G is any graceful graph of order m and size m - 1, both the corona G nK1 and the join G + nK1 are graceful. We discuss the gracefulness of several families thus obtained, like the coronas Cn mK1 and Kn mK1, enlarging considerably the known graceful families. We also include Skolem graceful labelings to study the gracefulness of the friendship graph K1 mP2. In addition, we prove that all hairy cycles are graceful . An hairy cycle is the graph obtained from a cycle by appending stars at some or all of its vertices. Thus coming closer to the conjecture of the gracefulness of unicyclic graphs posed by Truszczýnski.
Chapter 4 is dedicated to the gracefulness of disconnected graphs. The interest of this situation lies in the fact that disconnected
