Abstract.
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 v 1, …, v n such that v i∼H v j implies |i−j|≤b.¶ This conjecture has been proved in [1]. Answering a question of E. Szemerédi [6] 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.
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Received: May 20, 1998 Revised: September 1, 1999
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Abbasi, S. How Tight Is the Bollobás-Komlós Conjecture?. Graphs Comb 16, 129–137 (2000). https://doi.org/10.1007/PL00021175
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DOI: https://doi.org/10.1007/PL00021175