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Kneser graphs (Definition)

Let $ k,n$ be positive integers with $ k\leq n$. The Kneser graph $ K_{n:k}$ has as its vertex set the $ n\choose k$ $ k$-element subsets of $ \{1,2,\dots,n\}$. Two vertices are adjacent if and only if they correspond to disjoint subsets.

The graph $ K_{n:1}$ is the complete graph on $ n$ vertices. Another well-known Kneser graph is $ K_{5:2}$, better known as the Petersen graph, which is shown in figure 1. The Petersen graph is often a counterexample to graph-theoretical conjectures.

Figure 1: The Petersen graph
\scalebox{.5}{\includegraphics{PetersenGraph}}



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Also defines:  Petersen graph
Keywords:  Kneser graph, Petersen graph
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Cross-references: conjectures, counterexample, complete graph, graph, disjoint, adjacent, subsets, vertex, integers, positive
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This is version 2 of Kneser graphs, born on 2004-03-27, modified 2004-03-27.
Object id is 5732, canonical name is KneserGraphs.
Accessed 4283 times total.

Classification:
AMS MSC: 05C99 (Combinatorics :: Graph theory :: Miscellaneous)
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