# Schedule for: 18w5029 - Crossing Numbers: Theory and Applications

Arriving in Banff, Alberta on Sunday, October 21 and departing Friday October 26, 2018

Sunday, October 21 | |
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16:00 - 17:30 | Check-in begins at 16:00 on Sunday and is open 24 hours (Front Desk - Professional Development Centre) |

17:30 - 19:30 |
Dinner ↓ A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building. (Vistas Dining Room) |

20:00 - 22:00 | Informal gathering (Corbett Hall Lounge (CH 2110)) |

Monday, October 22 | |
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07:00 - 08:45 |
Breakfast ↓ Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building. (Vistas Dining Room) |

08:45 - 09:00 |
Introduction and Welcome by BIRS Staff ↓ A brief introduction to BIRS with important logistical information, technology instruction, and opportunity for participants to ask questions. (TCPL 201) |

09:00 - 10:00 |
Problem proposals ↓ The organizers will moderate a problem session where the main problems discussed during the week will be outlined and working groups formed. (TCPL 202) |

10:00 - 10:30 | Coffee Break (TCPL Foyer) |

10:30 - 11:30 | Problem proposals II (TCPL 202) |

11:30 - 13:00 |
Lunch ↓ Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building. (Vistas Dining Room) |

13:00 - 14:00 |
Guided Tour of The Banff Centre ↓ Meet in the Corbett Hall Lounge for a guided tour of The Banff Centre campus. (Corbett Hall Lounge (CH 2110)) |

14:00 - 14:15 | Free time (Courbet Hall) |

14:15 - 14:30 |
Group Photo ↓ Meet in foyer of TCPL to participate in the BIRS group photo. The photograph will be taken outdoors, so dress appropriately for the weather. Please don't be late, or you might not be in the official group photo! (Courbet Hall) |

14:30 - 15:00 |
Laszlo Szekely: Using block designs in crossing number bounds ↓ The {\em crossing number} $\Cr(G)$ of a graph $G=(V,E)$ is the smallest number of edge crossings over all drawings of $G$ in the plane. For any $k\ge 1$, the {\em $k$-planar crossing number} of $G$, $\Cr_k(G)$, is defined as the minimum of $\Cr(G_1)+\Cr(G_2)+\ldots+\Cr(G_{k})$ over all graphs $G_1, G_2,\ldots, G_{k}$ with $\cup_{i=1}^{k}G_i=G$. Pach et al. [\emph{Computational Geometry: Theory and Applications} {\bf 68} 2--6, (2018)] showed that for every $k\ge 1$, we have $\Cr_k(G)\le
\left(\frac{2}{k^2}-\frac1{k^3}\right)\Cr(G)$ and that this bound does not remain true if we replace the constant $\frac{2}{k^2}-\frac1{k^3}$ by any number smaller than $\frac1{k^2}$. We improve the upper bound to $\frac{1}{k^2}(1+o(1))$ as $k\rightarrow \infty$. For the class of bipartite graphs, we show that the best constant is exactly $\frac{1}{k^2}$ for every $k$. This result needed the existence of a variant of the midrange crossing constant for the class of bipartite graphs.
These results extend to the rectilinear variant of the $k$-planar crossing number.
This is joint work with {John Asplund}, {\'Eva Czabarka}, {Gregory Clark}, {Garner Cochran}, {Arran Hamm}, {Gwen Spencer}, {Libby Taylor} and {Zhiyu Wang}, originating from a Mathematics Research Communities program. (TCPL 202) |

15:00 - 15:30 | Coffee Break (TCPL Foyer) |

15:30 - 17:00 | Work in groups (TCPL 202) |

17:00 - 17:30 | Progress reports (TCPL 202) |

17:30 - 19:30 |
Dinner ↓ A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building. (Vistas Dining Room) |

Tuesday, October 23 | |
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07:00 - 09:00 | Breakfast (Vistas Dining Room) |

09:00 - 10:00 | Marcus Schaefer: Crossing Numbers: Some Open Questions (TCPL 201) |

10:00 - 10:30 | Coffee Break (TCPL Foyer) |

10:30 - 11:45 | Working in groups (TCPL 201) |

11:30 - 13:30 | Lunch (Vistas Dining Room) |

14:00 - 15:00 | Working in groups (TCPL 202) |

15:00 - 15:30 | Coffee Break (TCPL Foyer) |

15:30 - 17:00 | Working in groups (TCPL 202) |

17:00 - 17:30 | Progress reports (TCPL 202) |

17:30 - 19:30 | Dinner (Vistas Dining Room) |

Wednesday, October 24 | |
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07:00 - 09:00 | Breakfast (Vistas Dining Room) |

09:00 - 09:30 | Petr Hlineny: Crossing-critical graphs: Introduction (TCPL 201) |

09:30 - 10:00 | Zdenek Dvorak: Structure and generation of crossing-critical graphs (TCPL 201) |

10:00 - 10:30 | Coffee Break (TCPL Foyer) |

10:30 - 11:30 | Working in groups (TCPL 202) |

11:30 - 13:30 | Lunch (Vistas Dining Room) |

13:30 - 17:30 | Free Afternoon (Banff National Park) |

17:30 - 19:30 | Dinner (Vistas Dining Room) |

Thursday, October 25 | |
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07:00 - 09:00 | Breakfast (Vistas Dining Room) |

09:00 - 09:30 | Markus Chimani: Crossing number of random graphs (TCPL 201) |

09:30 - 10:00 | Drago Bokal: 2-crossing-critical graphs as lego bricks for inviting kids to graph theory (TCPL 201) |

10:00 - 10:30 | Coffee Break (TCPL Foyer) |

10:30 - 11:45 | Working in groups (TCPL 201) |

11:30 - 13:30 | Lunch (Vistas Dining Room) |

14:00 - 15:00 | Working in groups (TCPL 202) |

15:00 - 15:30 | Coffee Break (TCPL Foyer) |

15:30 - 17:00 | Working in groups (TCPL 202) |

17:00 - 17:45 | Progress reports (TCPL 202) |

17:30 - 19:30 | Dinner (Vistas Dining Room) |

Friday, October 26 | |
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07:00 - 09:00 | Breakfast (Vistas Dining Room) |

09:00 - 10:00 | Working in groups (TCPL 202) |

10:00 - 10:30 | Coffee Break (TCPL Foyer) |

10:30 - 11:30 | Progress reports (TCPL 202) |

11:30 - 12:00 |
Checkout by Noon ↓ 5-day workshop participants are welcome to use BIRS facilities (BIRS Coffee Lounge, TCPL and Reading Room) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 12 noon. (Front Desk - Professional Development Centre) |

12:00 - 13:30 | Lunch from 11:30 to 13:30 (Vistas Dining Room) |