### Abstract

Our main result implies the following easily formulated statement. The set of edges E of every finite bipartite graph can be split into poly(log | E |) subsets so that all the resulting bipartite graphs are almost regular. The latter means that the ratio between the maximal and minimal non-zero degree of the left nodes is bounded by a constant and the same condition holds for the right nodes. Stated differently, every finite 2-dimensional set S ⊂ N^{2} can be partitioned into poly (log | S |) parts so that in every part the ratio between the maximal size and the minimal size of non-empty horizontal section is bounded by a constant and the same condition holds for vertical sections. We prove a similar statement for n-dimensional sets for any n and show how it can be used to relate information inequalities for Shannon entropy of random variables to inequalities between sizes of sections and their projections of multi-dimensional finite sets.

Original language | English (US) |
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Pages (from-to) | 134-144 |

Number of pages | 11 |

Journal | European Journal of Combinatorics |

Volume | 28 |

Issue number | 1 |

DOIs | |

State | Published - Jan 2007 |

Externally published | Yes |

### All Science Journal Classification (ASJC) codes

- Discrete Mathematics and Combinatorics

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## Cite this

*European Journal of Combinatorics*,

*28*(1), 134-144. https://doi.org/10.1016/j.ejc.2005.08.002