The mathematical discipline of topological combinatorics is the application of topological and algebraic topological methods to solving problems in combinatorics.
In 1978 the situation was reversed — methods from algebraic topology were used to solve a problem in combinatorics – when László Lovász proved the Kneser conjecture, thus beginning the new study of topological combinatorics. Lovász's proof used the Borsuk–Ulam theorem and this theorem retains a prominent role in this new field. This theorem has many equivalent versions and analogs and has been used in the study of fair division problems.
In another application of homological methods to graph theory Lovász proved both the undirected and directed versions of a conjecture of András Frank: Given a k-connected graph G, k points , and k positive integers that sum up to , there exists a partition of such that , , and spans a connected subgraph.
In 1987 the necklace splitting problem was solved by Noga Alon using the Borsuk–Ulam theorem. It has also been used to study complexity problems in linear decision tree algorithms and the Aanderaa–Karp–Rosenberg conjecture. Other areas include topology of partially ordered sets and bruhat orders.
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|url=(help), Southampton, Hampshire: European Mathematical Society, pp. 16–19, retrieved 2008-07-29.
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