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Embedding Spanning Bipartite Graphs of Small Bandwidth

Published online by Cambridge University Press:  03 October 2012

FIACHRA KNOX  and
ANDREW TREGLOWN
FIACHRA KNOX
Affiliation:
School of Mathematics, University of Birmingham, Birmingham B15 2TT, UK (e-mail: knoxf@maths.bham.ac.uk)
ANDREW TREGLOWN
Affiliation:
Faculty of Mathematics and Physics, Charles University, Malostranské Náměstí 25, 188 00 Prague, Czech Republic (e-mail: treglown@kam.mff.cuni.cz)
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Abstract

Böttcher, Schacht and Taraz (Math. Ann., 2009) gave a condition on the minimum degree of a graph G on n vertices that ensures G contains every r-chromatic graph H on n vertices of bounded degree and of bandwidth o(n), thereby proving a conjecture of Bollobás and Komlós (Combin. Probab. Comput., 1999). We strengthen this result in the case when H is bipartite. Indeed, we give an essentially best-possible condition on the degree sequence of a graph G on n vertices that forces G to contain every bipartite graph H on n vertices of bounded degree and of bandwidth o(n). This also implies an Ore-type result. In fact, we prove a much stronger result where the condition on G is relaxed to a certain robust expansion property. Our result also confirms the bipartite case of a conjecture of Balogh, Kostochka and Treglown concerning the degree sequence of a graph which forces a perfect H-packing.

Keywords

Primary 05C35Secondary 05C7005C7805D40

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Type
Paper
Information
Combinatorics, Probability and Computing , Volume 22 , Issue 1 , January 2013 , pp. 71 - 96
Copyright
Copyright © Cambridge University Press 2012

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