Abstrakt
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 -> A(v) to the vertices v is an element of V such that A(u) and A(v) are disjoint if and only if uv is an element of E. The smallest such k is called the Kneser rank of G and denoted by fK(neser) (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 0, i = 1; 2, such that G is an element of G (n; p) satisfies with high probability c(1)n/(log n) = k: The smallest k such that there exists a k-mindi ff erence representation of G is denoted by fmin (G). Balogh and Prince proved in 2009 that for every k there is a graph G with f(min) (G) >= k.
We prove that there are constants c ''(1); c ''(2) > 0 such that c ''(1)n /(log n) < f(min) (G) < c ''(2) n / (log n) holds for almost all bipartite graphs G on n + n vertices.