Kneser

Kneser is a German-origin surname. Notable people with the name include Martin Kneser, a German mathematician known for introducing Kneser graphs, and Helmut Kneser, another German mathematician recognized for work in topology and algebra.

In mathematics, the term Kneser appears prominently in graph theory and combinatorics. Kneser graphs, denoted KG(n,k), have as vertices the k-element subsets of an n-element set, with edges joining pairs of disjoint subsets. These graphs are central objects in the study of graph colorings and combinatorial topology, and they serve as a testbed for ideas about intersection and disjointness among families of sets.

The Kneser conjecture concerned the chromatic number of KG(n,k). It states that the chromatic number is n − 2k + 2 for n ≥ 2k. The conjecture was proved by László Lovász in 1978 using topological methods, a landmark result that helped establish a fruitful link between topology and combinatorics and inspired further developments in the field.

Generalizations and related concepts include generalized Kneser graphs, which extend the construction to broader families of subsets, and Schrijver graphs, which are vertex-critical subgraphs of Kneser graphs that preserve the same chromatic number while often having more favorable properties. The Kneser framework thus anchors a notable area of study at the intersection of combinatorics, topology, and graph theory.