On Several Graph Labellings

In1966, in order to settle Ringel’s conjecture, Rosa introduces the concept ofgraph labellings: A graph labelling is a mapping from the vertex set of a graph toa set of integers. By various constraints we have many types of graph labellings.Clearly, graph labelling is an important branch of graph theory since it can beapplied to a wide of scientifc areas. However, there are many new problems yieldingin attacking famous conjectures, such as Graceful Tree Conjecture, Strongly GracefulTree Conjecture, Odd-graceful Tree Conjecture, Felicitous Tree Conjecture and soon.Based on the known conclusions and results on graph labellings, my researchingmainly focus on the following aspects:Chapter One distributes a simple introduction to the development of graphtheory and graph labellings. Basic terminology and notation of graph theory aredefned, and the defnitions, conjectures and some results of graph labellings aregiven.Chapter Two works mainly on graceful labelling of graphs. We list some resultsin current researching graceful labelling of graphs, and try to show a summary ofstudying graceful labelling of graphs. We investigate particular graceful labellingsin order to fnd some regularities of graceful labellings that can be used to constructlarge scale of graceful graphs, and do the operation of moving edge principle forattacking Graceful Tree Conjecture.In Chapter Three, we, frst of all, introduce the study on odd-graceful labelingsof graphs. Based on the study of Gnanajothi, for solved the conjecture proposed byBarrientos: Every bipartite graph is odd-graceful, and obtained some results. Wesolve the odd-gracefulness of several particular classes of bipartite graphs. The Fourth Chapter is to study graph felicitous labellings on which a few ofresults has been obtained. We prove that some particular bipartite graphs admitfelicitous labellings, and furthermore, by out experiences on working this topic, wepropose a conjecture: Every bipartite graph admits felicitous labellings.