Graceful graphs

In this thesis, graceful labeling of graphs is discussed and important results obtained on this subject have been compiled from various sources. Graph labeling, which began to be studied in the mid-1960s, is the process of assigning integers to vertices, edges, or both vertices and edges of a graph under certain conditions. Apart from graceful labeling, many graph labeling methods are still being studied intensively by researchers. Graceful labeling is a special graph labeling of a graph on m edges in which the nodes are labeled with a subset of distinct nonnegative integers from 0 to m and the graph edges are labeled with the absolute differences between node values. If the resulting graph edge numbers run from 1 to m inclusive, the labeling is called a graceful labeling and the graph is said to be a graceful graph. In the study, after giving preliminary information about the graphs, firstly the definition and basic properties of graceful graphs are listed and some important results characterizing graceful graphs are presented. Afterward, graceful labeling of basic graph families such as complete graph, complete bipartite graph, cycle graph, Euler graph is discussed. Then, graceful labeling of cycle-related graphs and graceful labeling of Cartesian products of graphs are discussed. Next, the Ringel-Kotzig conjecture and graceful labeling of many special tree families are presented. Finally, it is examined whether the union of graphs is graceful and an application of graceful graphs is given.