# Cutting a graph into two dissimilar halves

P. Erdős, Mark Goldberg, János Pach, Joel Spencer

Research output: Contribution to journalArticle

20 Citations (Scopus)

### 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 121-131 11 Journal of Graph Theory 12 1 https://doi.org/10.1002/jgt.3190120113 Published - 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. / Erdős, P.; Goldberg, Mark; Pach, János; Spencer, Joel.

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

Research output: Contribution to journalArticle

Erdős, P, Goldberg, M, Pach, J & Spencer, J 1988, 'Cutting a graph into two dissimilar halves', Journal of Graph Theory, vol. 12, no. 1, pp. 121-131. https://doi.org/10.1002/jgt.3190120113
Erdős, P. ; 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.
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