Extremal Graph Theory
نویسنده
چکیده
Extremal graph theory is, broadly speaking, the study of relations between various graph invariants, such as order, size, connectivity, minimum/maximum degree, chromatic number, etc., and the values of these invariants that ensure that the graph has certain properties. Since the first major result by Turan in 1941, numerous mathematicians have contributed to make this a vibrant and deep subject. Among all its topics, graph coloring is the most applicable and widely studied. Typically, a graph models conflicts, and a good coloring ensures partitions into parts with no conflicts. See Bollobás [2] and Jensen and Toft [8] for an overview.
منابع مشابه
Essays in extremal combinatorics
We prove several results from different areas of extremal combinatorics, giving complete or partial solutions to a number of open problems. These results, coming from areas such as extremal graph theory, Ramsey theory and additive combinatorics, have been collected together because in each case the relevant proofs are quite short.
متن کاملExtremal infinite graph theory
We survey various aspects of infinite extremal graph theory and prove several new results. The lead role play the parameters connectivity and degree. This includes the end degree. Many open problems are suggested.
متن کاملRational exponents in extremal graph theory
Given a family of graphs H, the extremal number ex(n,H) is the largest m for which there exists a graph with n vertices and m edges containing no graph from the family H as a subgraph. We show that for every rational number r between 1 and 2, there is a family of graphs Hr such that ex(n,Hr) = Θ(n r). This solves a longstanding problem in the area of extremal graph theory.
متن کاملOn Some Interconnections between Combinatorial Optimization and Extremal Graph Theory
The uniting feature of combinatorial optimization and extremal graph theory is that in both areas one should find extrema of a function defined in most cases on a finite set. While in combinatorial optimization the point is in developing efficient algorithms and heuristics for solving specified types of problems, the extremal graph theory deals with finding bounds for various graph invariants u...
متن کاملShort proofs of some extremal results II
We prove several results from different areas of extremal combinatorics, including complete or partial solutions to a number of open problems. These results, coming mainly from extremal graph theory and Ramsey theory, have been collected together because in each case the relevant proofs are quite short.
متن کامل