The equivariant topology of stable Kneser graphs

The stable Kneser graph S G n , k , n ⩾ 1 , k ⩾ 0 , introduced by Schrijver (1978) [19], is a vertex critical graph with chromatic number k + 2 , its vertices are certain subsets of a set of cardinality m = 2 n + k . Björner and de Longueville (2003) [5] have shown that its box complex is homotopy equivalent to a sphere, Hom ( K 2 , S G n , k ) ≃ S k . The dihedral group D 2 m acts canonically on S G n , k , the group C 2 with 2 elements acts on K 2 . We almost determine the ( C 2 × D 2 m ) -homotopy type of Hom ( K 2 , S G n , k ) and use this to prove the following results. aphs S G 2 s , 4 are homotopy test graphs, i.e. for every graph H and r ⩾ 0 such that Hom ( S G 2 s , 4 , H ) is ( r − 1 ) -connected, the chromatic number χ ( H ) is at least r + 6 . { 0 , 1 , 2 , 4 , 8 } and n ⩾ N ( k ) then S G n , k is not a homotopy test graph, i.e. there are a graph G and an r ⩾ 1 such that Hom ( S G n , k , G ) is ( r − 1 ) -connected and χ ( G ) < r + k + 2 .