Periodic Reporting for period 2 - SSiGraph (Spanning Subgraphs in Graphs)
Reporting period: 2022-04-01 to 2023-09-30
The project aims to address a range of exciting and challenging extremal and probabilistic problems on spanning subgraphs in graphs from the Ryser-Brualdi-Stein conjecture to appearance threshold questions in random graphs.
A partial transversal in a Latin square of order n is a set of cells which share no column, row or symbol while a full transversal is a partial transversal with n cells. The Ryser-Brualdi-Stein conjecture from 1967 says that every Latin square of order n has a partial transversal with n-1 cells, and moreover a full transversal if n is odd.
The project has shown that if n is taken to be sufficiently large, then every Latin square of order n has a partial transversal with n-1 cells. This proves the conjecture for large even n, and improves the previous best known bound of n-O(log n/loglog n) cells.