2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/159729For each m>=1 and k>=2, we construct a graph G=(V,E) with ω(G)=m such that max_{1\leq i\leq k} ω(G[V_i])=m for arbitrary partition V=V_1\cup...\cup V_k, where ω(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.7 pages. Conjecture 3.1 in the first version has been solvedCombinatoricsNumber Theory05C30 (Primary); 11B25, 05C90, 05C20(Secondary)text