## Abstract

In this paper, we use semidefinite programming and representation theory to compute new lower bounds on the crossing number of the complete bipartite graph Km,n, extending a method from de Klerk et al. (SIAM J Discrete Math 20:189–202, 2006) and the subsequent reduction by De Klerk, Pasechnik and Schrijver (Math Prog Ser A and B 109:613–624, 2007). We exploit the full symmetry of the problem using a novel decomposition technique. This results in a full block-diagonalization of the underlying matrix algebra, which we use to improve bounds on several concrete instances. Our results imply that cr(K10,n) ≥ 4.87057n2 − 10n, cr(K11,n) ≥ 5.99939n2 − 12.5n, cr(K12,n) ≥ 7.25579n2 − 15n, cr(K13,n) ≥ 8.65675n2 − 18n for all n. The latter three bounds are computed using a new and well-performing relaxation of the original

semidefinite programming bound. This new relaxation is obtained by only requiring one small matrix block to be positive semidefinite.

semidefinite programming bound. This new relaxation is obtained by only requiring one small matrix block to be positive semidefinite.

Original language | English |
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Pages (from-to) | 693-715 |

Number of pages | 23 |

Journal | Mathematical Programming |

Volume | 207 |

Issue number | 1-2 |

DOIs | |

Publication status | Published - Sept 2024 |

## Keywords

- crossing numbers
- complete bipartite graph
- semidefinite programming
- symmetry reduction
- block-diagonalization