Spanning Subgraphs in Graphs

Informations projet

SSiGraph

N° de convention de subvention: 947978

DOI

10.3030/947978

Date de signature de la CE 12 Août 2020

Date de début 1 Janvier 2021

Date de fin 31 Decembre 2026

Financé au titre de

• EXCELLENT SCIENCE - European Research Council (ERC)

Coût total

€ 1 417 365,00

Contribution de l’UE

€ 1 417 365,00

Coordonné par

UNIVERSITY OF WARWICK
United Kingdom

Periodic Reporting for period 3 - SSiGraph (Spanning Subgraphs in Graphs)

Période du rapport: 2023-10-01 au 2025-06-30

Résumé du contexte et des objectifs généraux du projet

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.

Travail effectué depuis le début du projet jusqu’à la fin de la période considérée dans le rapport et principaux résultats atteints jusqu’à présent

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.

Progrès au-delà de l’état des connaissances et impact potentiel prévu (y compris l’impact socio-économique et les conséquences sociétales plus larges du projet jusqu’à présent)

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.