Clique numbers of graphs and irreducible exact m-covers of the integers

For each m⩾1m⩾1 and k⩾2k⩾2, we construct a graph G=(V,E)G=(V,E) with ω(G)=mω(G)=m such thatmax1⩽i⩽kω(G[Vi])=m for arbitrary partition {V1,…,Vk}{V1,…,Vk} of V  , where ω(G)ω(G) is the clique number of G   and G[Vi]G[Vi] is the induced subgraph of G   with the vertex set ViVi. Using this result, we show that for each m⩾2m⩾2 there exists an exact m  -cover of ZZ which is not the union of two 1-covers.