Kneser ranks of random graphs and minimum difference representations

Abstract

Every graph G=(V,E) is an induced subgraph of some Kneser graph of rank k, i.e., there is an assignment of (distinct) k-sets v↦Av to the vertices v∈V such that Au and Av are disjoint if and only if uv∈E. The smallest such k is called the Kneser rank of G and denoted by fKneser(G). As an application of a result of Frieze and Reed concerning the clique cover number of random graphs we show that for constant 0<p<1 there exist constants ci=ci(p)>0, i=1,2 such that with high probabilityc1n/(log⁡n)<fKneser(G)<c2n/(log⁡n). We apply this to other graph representations defined by Boros, Gurvich and Meshulam.A k-min-difference representation of a graph G is an assignment of a set Ai to each vertex i∈V(G) such that ij∈E(G)⇔min⁡{|Ai\\Aj|,|Aj\\Ai|}≥k. The smallest k such that there exists a k-min-difference representation of G is denoted by fmin(G). Balogh and Prince proved in 2009 that for every k there is a graph G with fmin(G)≥k. We prove that there are constants c1″,c2″>0 such that c1″n/(log⁡n)<fmin(G)<c2″n/(log⁡n) holds for almost all bipartite graphs G on n+n vertices.