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

Abstract

For each m ⩾ 1 and k ⩾ 2 , we construct a graph G = ( V , E ) with ω ( G ) = m such that max 1 ⩽ i ⩽ k ω ( G [ V i ] ) = m for arbitrary partition { V 1 , … , V k } of V, 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.