Introduction
Graphs are mathematical structures that capture the pairwise relationship of objects. Because graphs can represent pairwise relations, graph theory has applications in diverse fields including computer science, biochemistry, and electrical engineering.
One topic of interest in graph theory is the study of graph labelings. A graph is labeled with a function that takes each vertex and assigns it a number.
One topic of interest in graph theory is the study of graph labelings. A graph is labeled with a function that takes each vertex and assigns it a number.
The first type of labeling that was studied in depth was a graceful labeling by Rosa. A graph labeling is graceful when the difference between each adjacent vertex is unique for all edges modulo the number of edges. A famous unsolved problem relating to graceful labelings is the Graceful Tree Conjecture (also called the Ringel-Kotzig Conjecture), which says that all finite trees are graceful. This result has been proven for infinite graphs.
Since Rosa, over 2,000 papers on labelings have been written (see Gallian for an extensive survey). An example of one of these papers is (Graham, Sloane 1980), which showed that certain families of finite graphs, such as caterpillars, are harmonious. Our research focused on extending the definition of harmonious labelings to infinite graphs and then determining which families of infinite graphs are harmonious.
Since Rosa, over 2,000 papers on labelings have been written (see Gallian for an extensive survey). An example of one of these papers is (Graham, Sloane 1980), which showed that certain families of finite graphs, such as caterpillars, are harmonious. Our research focused on extending the definition of harmonious labelings to infinite graphs and then determining which families of infinite graphs are harmonious.