2008/04/06 by Hao Pan, Pan, Hao, Li-Lu Zhao +1
Mathematics · #05C20(Secondary) #05C30 (Primary) #05C90 #11B25 #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT) #math.CO #math.NT #msc:05C30 #msc:05C90 #msc:11B25
paper · pdf · doi:10.48550/arxiv.0804.0901
7 pages. Conjecture 3.1 in the first version has been solved
arxiv created 2008/04/26 · arxiv updated 2009/12/01
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.