Clique Numbers of Graphs and Irreducible Exact m-Covers of Z

For each m>=1 and k>=2, we construct a graph G=(V,E) with \omega(G)=m such that max_{1\leq i\leq k} \omega(G[V_i])=m for arbitrary partition V=V_1\cup...\cup V_k, where \omega(G) is the clique number of G and G[V_i] is the induced subgraph of G with the vertex set V_i. Using this result, we show that for each m>=2 there exists an exact m-cover of Z which is not the union of two 1-covers.