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Cutting a graph into two dissimilar halves
Author(s) -
Erdós Paul,
Goldberg Mark,
Pach János,
Spencer Joel
Publication year - 1988
Publication title -
journal of graph theory
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.164
H-Index - 54
eISSN - 1097-0118
pISSN - 0364-9024
DOI - 10.1002/jgt.3190120113
Subject(s) - combinatorics , mathematics , bipartite graph , vertex (graph theory) , graph , complete bipartite graph , discrete mathematics
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.