Chromatic number via Turán number

For a graph G and a family of graphs F, the general Kneser graph KG(G, F) is a graph with the vertex set consisting of all subgraphs of G isomorphic to some member of F and two vertices are adjacent if their corresponding subgraphs are edge disjoint. In this paper, we introduce some generalizations of Turán number of graphs.

In view of these generalizations, we give some lower and upper bounds for the chromatic number of general Kneser graphs KG(G, F). Using these bounds, we determine the chromatic number of some family of general Kneser graphs KG(G, F) in terms of generalized Turán number of graphs. In particular, we determine the chromatic number of every Kneser multigraph KG(G, F) where G is a multigraph each of whose edges has the multiplicity at least 2 and F is an arbitrary family of simple graphs.

Moreover, the chromatic number of general Kneser graph KG(G, F) is exactly determined where G is a dense graph and F = {K1,2}