Cutting a graph into two dissimilar halves

Paul Erdós, Mark Goldberg, János Pach, Joel Spencer

Research output: Contribution to journalArticlepeer-review


Given a graph G and a subset S of the vertex set of G, the discrepancy of S is defined as the difference between the actual and expected numbers of the edges in the subgraph induced on S. We show that for every graph with n vertices and e edges, n < e < n(n − 1)/4, there is an n/2‐element subset with the discrepancy of the order of magnitude of \documentclass{article}\pagestyle{empty}\begin{document}$\sqrt {ne}$\end{document} For graphs with fewer than n edges, we calculate the asymptotics for the maximum guaranteed discrepancy of an n/2‐element subset. We also introduce a new notion called “bipartite discrepancy” and discuss related results and open problems.

Original languageEnglish (US)
Pages (from-to)121-131
Number of pages11
JournalJournal of Graph Theory
Issue number1
StatePublished - 1988

ASJC Scopus subject areas

  • Discrete Mathematics and Combinatorics
  • Geometry and Topology


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