Improvisations on the Hall marriage theorem: Completing Latin squares and Sudokus

Seminar
Speaker
Eli Shamir (Hebrew University, Jerusalem)
Date
06/03/2016 - 13:00 - 12:00Add to Calendar 2016-03-06 12:00:00 2016-03-06 13:00:00 Improvisations on the Hall marriage theorem: Completing Latin squares and Sudokus The key concept of our discussion is that of a perfect matching (PM) in a bipartite graph. The expansion condition in Hall's marriage theorem can be extended to an unbiased 2-sided one. This enables an alternative (and simpler) proof of Evans' (proven) Conjecture: A partial nxn Latin square with n-1 dictated entries admits a completion to a full Latin square.  PMs are used to successively fill the square by rows, columns or diagonals. Latin square tables correspond to quasi-groups; the ones corresponding to groups are only a tiny fraction of them, as n grows. However, for Sudoku tables of order mnxmn, the completion (say by diagonals) usually fails, even if there are no dictated entries, unless they are conjugates of a twisted product of two groups, of orders n and m. An open problem for sudoku lovers: Is there a sudoku square (of any order) which is not a conjugate of a twisted product of groups? No prior knowledge needed.   Department room אוניברסיטת בר-אילן - Department of Mathematics mathoffice@math.biu.ac.il Asia/Jerusalem public
Place
Department room
Abstract

The key concept of our discussion is that of a perfect matching (PM) in a bipartite graph. The expansion condition in Hall's marriage theorem can be extended to an unbiased 2-sided one. This enables an alternative (and simpler) proof of Evans' (proven) Conjecture:

A partial nxn Latin square with n-1 dictated entries admits a completion to a full Latin square. 

PMs are used to successively fill the square by rows, columns or diagonals. Latin square tables correspond to quasi-groups; the ones corresponding to groups are only a tiny fraction of them, as n grows. However, for Sudoku tables of order mnxmn, the completion (say by diagonals) usually fails, even if there are no dictated entries, unless they are conjugates of a twisted product of two groups, of orders n and m.

An open problem for sudoku lovers: Is there a sudoku square (of any order) which is not a conjugate of a twisted product of groups?

No prior knowledge needed.

 

Last Updated Date : 09/03/2016