# Graph Theory

# Graphs

For the purposes of the research I have done, I use the following definitions. These constitute a simple, undirected graph.

Graph: A graph is a pair of sets. A vertex set containing labels for vertices. An edge set containing unordered pairs of distinct vertices.

Graph Labeling: A graph labeling is any assignment of labels to the vertex set, the edge set, or both sets of a graph.

## Classical Labelings

Harmonious Labeling: A vertex labeling from the group of integers modulo n that induces an edge labeling from the same group. The edge labeling is defined as the modular sum of the two incident vertices.

Harmonious Graph: Any graph for which there exists a harmonious labeling that repeats no edge label.

M-Harmonious Labeling: A vertex labeling from the group of integers modulo n relatively prime to n that induces an edge labeling from the same group. The edge labeling is defined as the modular sum of the two incident vertices.

M-Harmonious Graph: Any graph for which there exists a M-harmonious labeling that repeats no edge label

Γ-Harmonious Labeling: A vertex labeling from the cartesian product of groups of integers modulo n that induces an edge labeling from the same group. The edge labeling is defined as the direct product of the two incident vertices.

Γ-Harmonious Graph: Any graph for which there exists a Γ-harmonious labeling that repeats no edge label

Power Domination: A graph coloring game in which the following two steps occur with a given set of vertices.

(Domination Step) Color blue each vertex in, and all adjacent vertices to vertices in the given set.

(Zero Forcing Step) While there exists a blue vertex with exactly one uncolored neighbor, color that neighbor blue.

Power Dominating Set: Any set of vertices for which the power domination process colors the entire graph.

Power Domination Number: The smallest number of vertices required for a power dominating set of a given graph.

Ramsey Theory in relation to graph theory is the search of structure within all possible red-blue edge colorings of a given graph. I take this concept in a novel way by looking at building the set of all red-blue edge colorings of a graph in a "minimal" manner, being certain that no coloring is repeated.