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.