is a key result in . It establishes the maximum number of edges a graph can have without containing certain complete subgraphs, linking to structural properties.
This theorem forms the basis for many other results in the field. It introduces important concepts like , edge density, and Turán graphs, which are crucial for understanding graph structure and properties.
Turán's Theorem and Extremal Graph Theory
Turán's theorem in graph theory
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Turán's theorem fundamental result in extremal graph theory provides upper bound on edge count in graphs without complete subgraphs
Extremal graph theory studies maximum or minimum graphs satisfying certain properties focuses on relationships between graph parameters and structural properties
Turán's theorem establishes connection between edge density and presence of complete subgraphs serves as foundation for many other results in extremal graph theory
Key concepts include clique number (size of largest complete subgraph), edge density (ratio of edges to maximum possible edges), and (extremal graph achieving maximum edges without containing specific complete subgraph)
Maximum edges without subgraphs
Turán's theorem states for graph G with n vertices and no Kr+1 subgraph, maximum number of edges is e(G)≤2rr−1n2
Applies to triangle-free graphs (r = 2): e(G)≤4n2 and K4-free graphs (r = 3): e(G)≤3n2
generalizes Turán's result to forbidden subgraphs other than complete graphs
Techniques for determining maximum edges include induction on vertex count, averaging arguments, and extremal graph constructions
Properties of Turán graphs
Turán graph complete with n vertices divided as equally as possible
Construction process:
Divide n vertices into r parts
Connect all vertices between different parts
Ensure size difference between parts is at most 1
Properties include (contains maximum edges without Kr+1 subgraph), symmetry (vertex-transitive), and degree distribution (vertices have degree rr−1n or rr−1n−1)
Examples include T(n,2) ( with parts of size ⌊2n⌋ and ⌈2n⌉) and T(n,3) ( with parts differing by at most 1 in size)
Applications of Turán's theorem
Problem-solving strategies involve identifying forbidden subgraph, applying Turán's theorem for edge upper bound, and constructing extremal graph to show bound tightness
Extensions include (forbidding multiple subgraphs), for hypergraphs, and (using eigenvalues to bound subgraph occurrences)
Applications to other areas include (connections between Turán's theorem and Ramsey numbers) and (using Turán's theorem to prove existence of certain graphs)
Advanced techniques involve (characterizing graphs close to extremal), (counting forbidden subgraphs in dense graphs), and (computer-assisted proofs in extremal graph theory)