On the bandwidth conjecture for 3-colourable graphs

Authors:
Julia Böttcher;Mathias Schacht;Anusch Taraz
Affiliations:
Zentrum Mathematik, Technische Universität München, Boltzmannstraße, Garching bei München, Germany;Institut für Informatik, Humboldt-Universität zu Berlin, Unter den Linden, Berlin, Germany;Zentrum Mathematik, Technische Universität München, Boltzmannstraße, Garching bei München, Germany
Venue:
SODA '07 Proceedings of the eighteenth annual ACM-SIAM symposium on Discrete algorithms
Year:
2007

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Abstract

A conjecture by Bollobás and Komlós states that for every γ 0 and integers r ≥ 2 and Δ, there exists β 0 such that for sufficiently large n the following holds: If G is a graph on n vertices with minimum degree at least (r-1/r+γ)n and H is an r-chromatic graph on n vertices with bandwidth at most Βn and maximum degree at most Δ, then G contains a copy of H. This conjecture generalises several results concerning sufficient degree conditions for the containment of spanning subgraphs. We prove the conjecture for the case r = 3. Our proof yields a polynomial time algorithm for embedding H into G if H is given together with a 3-colouring and vertex labelling respecting the bandwidth bound.