The neighborhood complexes of almost s-stable Kneser graphs

In 1978, László Lovász proved the famous Kneser Conjecture - concerning the chromatic number of the Kneser graphs KGm,n - introducing the neighborhood complex. In the same year, Alexander Schrijver defined certain induced subgraphs of KGm,n - called the stable Kneser graphs SGm,n - and showed that they are vertex-critical. Schrijver used another, Bárány's method to obtain the chromatic number of the stable Kneser graphs. Almost 25 years later, in 2002, Anders Björner and Mark de Longueville studied the neighborhood complex of SGm,n, and determined the homotopy type of them. Frédéric Meunier generalized Schrijver's construction and formulated the conjecture on the chromatic number of the s-stable and almost s-stable Kneser graphs. We shall determine the homotopy type of the neighborhood complex of the almost s-stable Kneser graphs. In conjunction with Lovász's topological bound on the chromatic number, we give the chromatic number of these graphs, which was recently determined by Chen using other methods.