### Abstract

We show the existence of a non-constant gap between the communication complexity of a function and the logarithm of the rank of its input matrix. We consider the following problem: each of two players gets a perfect matching between two n-element sets of vertices. Their goal is to decide whether or not the union of the two matcliings forms a Hamiltonian cycle. We prove: 1) The rank of the input matrix over the reals for this problem is 2^{O(n)}. 2) The non-deterministic communication complexity of the problem is Ω(nloglog n). Our result also supplies a superpolynomial gap between the chromatic number of a graph and the rank of its adjacency matrix. Another conclusion from the second result is an Ω(nloglog n) lower bound for the graph connectivity problem in the non-deterministic case. We make use of the theory of group representations for the first result. The second result is proved by an information theoretic argument.

Original language | English (US) |
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Pages (from-to) | 567-588 |

Number of pages | 22 |

Journal | Combinatorica |

Volume | 15 |

Issue number | 4 |

DOIs | |

State | Published - Dec 1 1995 |

Externally published | Yes |

### All Science Journal Classification (ASJC) codes

- Discrete Mathematics and Combinatorics
- Computational Mathematics

### Keywords

- Mathematics Subject Classification (1991): 68Q15, 05C50, 68R10, 05C15

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## Cite this

*Combinatorica*,

*15*(4), 567-588. https://doi.org/10.1007/BF01192528