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

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

2008-04-06

arxiv.org/abs/0804.0901v3

Mathematics

Combinatorics

7 pages. Conjecture 3.1 in the first version has been solved

Scientific paper

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.

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