The Maximum Size of an Edge Cut and Graph Homomorphisms

For a graph G, let b(G)=max|D|: Dis an edge cut of G . For graphs G and H, a map Ψ: V(G)→V(H) is a graph homomorphism if for each e=uv∈E(G), Ψ(u)Ψ(v)∈E(H). In 1979, Erdös proved by probabilistic methods that for p ≥ 2 with if there is a graph homomorphism from G onto K_p then b(G)≥f(p)|E(G)| In this paper, we obtained the best possible lower bounds of b(G) for graphs G with a graph homomorphism onto a Kneser graph or a circulant graph and we characterized the graphs G reaching the lower bounds when G is an edge maximal graph with a graph homomorphism onto a complete graph, or onto an odd cycle