## Abstract

For a graph G = (V, E), let bp(G) denote the minimum number of pairwise edge disjoint complete bipartite subgraphs of G so that each edge of G belongs to exactly one of them. It is easy to see that for every graph G, bp(G) ≤ n − α(G), where α(G) is the maximum size of an independent set of G. Erdős conjectured in the 80s that for almost every graph G equality holds, that is that for the random graph G(n, 0.5), bp(G) = n − α(G) with high probability, that is with probability that tends to 1 as n tends to infinity. The first author showed that this is slightly false, proving that for most values of n tending to infinity and for G = G(n, 0.5), bp(G) ≤ n − α(G) − 1 with high probability. We prove a stronger bound: there exists an absolute constant c > 0 so that bp(G) ≤ n − (1 + c)α(G) with high probability.

Original language | English (US) |
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Pages (from-to) | 45-52 |

Number of pages | 8 |

Journal | Journal of Graph Theory |

Volume | 84 |

Issue number | 1 |

DOIs | |

State | Published - Jan 1 2017 |

Externally published | Yes |

## All Science Journal Classification (ASJC) codes

- Geometry and Topology
- Discrete Mathematics and Combinatorics

## Keywords

- bipartite decomposition
- random graph