Extremal graph theory

Extremal graph theory is a branch of combinatorics, itself an area of mathematics, that lies at the intersection of extremal combinatorics and graph theory. In essence, extremal graph theory studies how global properties of a graph influence local substructure. Results in extremal graph theory deal with quantitative connections between various graph properties, both global (such as the number of vertices and edges) and local (such as the existence of specific subgraphs), and problems in extremal graph theory can often be formulated as optimization problems: how big or small a parameter of a graph can be, given some constraints that the graph has to satisfy? A graph that is an optimal solution to such an optimization problem is called an extremal graph, and extremal graphs are important objects of study in extremal graph theory. Extremal graph theory is closely related to fields such as Ramsey theory, spectral graph theory, computational complexity theory, and additive combinatorics, and frequently employs the probabilistic method. Mantel's Theorem (1907) and Turán's Theorem (1941) were some of the first milestones in the study of extremal graph theory. In particular, Turán's theorem would later on become a motivation for the finding of results such as the Erdős–Stone theorem (1946). This result is surprising because it connects the chromatic number with the maximal number of edges in an -free graph. An alternative proof of Erdős–Stone was given in 1975, and utilised the Szemerédi regularity lemma, an essential technique in the resolution of extremal graph theory problems. Graph coloring A proper (vertex) coloring of a graph is a coloring of the vertices of such that no two adjacent vertices have the same color. The minimum number of colors needed to properly color is called the chromatic number of , denoted . Determining the chromatic number of specific graphs is a fundamental question in extremal graph theory, because many problems in the area and related areas can be formulated in terms of graph coloring.