Clique Numbers of Graphs and Irreducible Exact m-Covers of Z
Abstract
For each m>=1 and k>=2, we construct a graph G=(V,E) with ω(G)=m such that max1≤ i≤ k ω(G[Vi])=m for arbitrary partition V=V1... Vk, where ω(G) is the clique number of G and G[Vi] is the induced subgraph of G with the vertex set Vi. 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.
0