Combinatorica

, Volume 36, Issue 5, pp 537–556 | Cite as

On judicious bipartitions of graphs

Article

Abstract

For a positive integer m, let f(m) be the maximum value t such that any graph with m edges has a bipartite subgraph of size at least t, and let g(m) be the minimum value s such that for any graph G with m edges there exists a bipartition V (G)=V1V2 such that G has at most s edges with both incident vertices in Vi. Alon proved that the limsup of \(f\left( m \right) - \left( {m/2 + \sqrt {m/8} } \right)\) tends to infinity as m tends to infinity, establishing a conjecture of Erdős. Bollobás and Scott proposed the following judicious version of Erdős' conjecture: the limsup of \(m/4 + \left( {\sqrt {m/32} - g(m)} \right)\) tends to infinity as m tends to infinity. In this paper, we confirm this conjecture. Moreover, we extend this conjecture to k-partitions for all even integers k. On the other hand, we generalize Alon's result to multi-partitions, which should be useful for generalizing the above Bollobás-Scott conjecture to k-partitions for odd integers k.

Mathematics Subject Classification (2000)

05C35 

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Copyright information

© János Bolyai Mathematical Society and Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.School of Mathematical SciencesUniversity of Science and Technology of ChinaHefei, AnhuiChina
  2. 2.School of MathematicsGeorgia Institute of Technology AtlantaGeorgiaUSA

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