74 research outputs found
The equivariant topology of stable Kneser graphs
The stable Kneser graph , , , introduced by Schrijver
\cite{schrijver}, is a vertex critical graph with chromatic number , its
vertices are certain subsets of a set of cardinality . Bj\"orner and de
Longueville \cite{anders-mark} have shown that its box complex is homotopy
equivalent to a sphere, \Hom(K_2,SG_{n,k})\homot\Sphere^k. The dihedral group
acts canonically on , the group with 2 elements acts
on . We almost determine the -homotopy type of
\Hom(K_2,SG_{n,k}) and use this to prove the following results. The graphs
are homotopy test graphs, i.e. for every graph and such
that \Hom(SG_{2s,4},H) is -connected, the chromatic number
is at least . If and then
is not a homotopy test graph, i.e.\ there are a graph and an such
that \Hom(SG_{n,k}, G) is -connected and .Comment: 34 pp
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