Cutting a graph into two dissimilar halves

Paul Erdós, Mark Goldberg, János Pach, Joel Spencer

Research output: Contribution to journalArticle

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 languageEnglish (US)
Pages (from-to)121-131
Number of pages11
JournalJournal of Graph Theory
Volume12
Issue number1
DOIs
StatePublished - 1988

Fingerprint

Discrepancy
Graph in graph theory
Subset
Induced Subgraph
Open Problems
Calculate
Vertex of a graph

ASJC Scopus subject areas

  • Geometry and Topology

Cite this

Cutting a graph into two dissimilar halves. / Erdós, Paul; Goldberg, Mark; Pach, János; Spencer, Joel.

In: Journal of Graph Theory, Vol. 12, No. 1, 1988, p. 121-131.

Research output: Contribution to journalArticle

Erdós, Paul ; Goldberg, Mark ; Pach, János ; Spencer, Joel. / Cutting a graph into two dissimilar halves. In: Journal of Graph Theory. 1988 ; Vol. 12, No. 1. pp. 121-131.
@article{8a087094c53d42d1a03bd41c9eba9175,
title = "Cutting a graph into two dissimilar halves",
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.",
author = "Paul Erd{\'o}s and Mark Goldberg and J{\'a}nos Pach and Joel Spencer",
year = "1988",
doi = "10.1002/jgt.3190120113",
language = "English (US)",
volume = "12",
pages = "121--131",
journal = "Journal of Graph Theory",
issn = "0364-9024",
publisher = "Wiley-Liss Inc.",
number = "1",

}

TY - JOUR

T1 - Cutting a graph into two dissimilar halves

AU - Erdós, Paul

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 -