### Abstract

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 language | English |
---|---|

Pages (from-to) | 121-131 |

Number of pages | 11 |

Journal | Journal of Graph Theory |

Volume | 12 |

Issue number | 1 |

DOIs | |

Publication status | Published - 1988 |

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### ASJC Scopus subject areas

- Geometry and Topology

### Cite this

*Journal of Graph Theory*,

*12*(1), 121-131. https://doi.org/10.1002/jgt.3190120113

**Cutting a graph into two dissimilar halves.** / Erdős, P.; Goldberg, Mark; Pach, János; Spencer, Joel.

Research output: Contribution to journal › Article

*Journal of Graph Theory*, vol. 12, no. 1, pp. 121-131. https://doi.org/10.1002/jgt.3190120113

}

TY - JOUR

T1 - Cutting a graph into two dissimilar halves

AU - Erdős, P.

AU - Goldberg, Mark

AU - Pach, János

AU - Spencer, Joel

PY - 1988

Y1 - 1988

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=84986529368&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84986529368&partnerID=8YFLogxK

U2 - 10.1002/jgt.3190120113

DO - 10.1002/jgt.3190120113

M3 - Article

VL - 12

SP - 121

EP - 131

JO - Journal of Graph Theory

JF - Journal of Graph Theory

SN - 0364-9024

IS - 1

ER -