How Tight Is the Bollobás-Komlós Conjecture?
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The bipartite case of the Bollobás and Komlós conjecture states that for every Δ0, γ>0 there is an α=α(Δ0, γ) >0 such that the following statement holds: If G is any graph with minimum degree at least \(\) then G contains as subgraphs all n vertex bipartite graphs, H, satisfying¶
¶Here b(H), the bandwidth of H, is the smallest b such that the vertices of H can be ordered as v1, …, vn such that vi∼Hvj implies |i−j|≤b.¶ This conjecture has been proved in . Answering a question of E. Szemerédi  we show that this conjecture is tight in the sense that as γ→0 then α→0. More precisely, we show that for any \(\) there is a Δ0 such that that α(Δ0, γ)≤4 γ.
KeywordsBipartite Graph Minimum Degree Bipartite Case
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