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# Spanning Subgraphs in Graphs

## Periodic Reporting for period 2 - SSiGraph (Spanning Subgraphs in Graphs)

Reporting period: 2022-04-01 to 2023-09-30

Graph Theory is a highly active area of Combinatorics with strong links to fields such as Optimisation and Theoretical Computer Science. A fundamental meta-problem in Graph Theory is the following: given a graph H, what conditions guarantee that a graph G contains a copy of H as a subgraph? This is particularly important when H has the same number of vertices as G, where we say that such a copy of H spans G. Many significant and particularly challenging problems in Graph Theory concern spanning subgraphs, where there is no room for manoeuvre as the copy of H must fit exactly into G.

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.
The project has made progress on a range of problems. Most notable is the work on the Ryser-Brualdi-Stein conjecture on Latin squares, whose rigorous study originates with Euler in the 18th century. A Latin square of order n is an n by n grid filled with n different symbols so that no two of the same symbols appear in any row or column --- an example is the underlying grid of a sudoku square which is a Latin square of order 9. Latin squares occur naturally as the multiplication tables of finite groups and are connected to areas including permutations and error correcting codes.

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.
With the work on the Ryser-Brualdi-Stein conjecture, the project has pushed the state-of-the-art, introducing several key new techniques very different to the previous work on this conjecture and related areas. The project continues to develop these, and other methods introduced, alongside a range of work on exciting problems.