Spanning Subgraphs in Graphs

Projektinformationen

SSiGraph

ID Finanzhilfevereinbarung: 947978

DOI

10.3030/947978

EK-Unterschriftsdatum 12 August 2020

Startdatum 1 Januar 2021

Enddatum 31 Dezember 2026

Finanziert unter

• EXCELLENT SCIENCE - European Research Council (ERC)

Gesamtkosten

€ 1 417 365,00

EU-Beitrag

€ 1 417 365,00

Koordiniert durch

UNIVERSITY OF WARWICK
United Kingdom

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

Berichtszeitraum: 2023-10-01 bis 2025-06-30

Zusammenfassung vom Kontext und den Gesamtzielen des Projekts

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.

Arbeit, die ab Beginn des Projekts bis zum Ende des durch den Bericht erfassten Berichtszeitraums geleistet wurde, und die wichtigsten bis dahin erzielten Ergebnisse

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.

Fortschritte, die über den aktuellen Stand der Technik hinausgehen und voraussichtliche potenzielle Auswirkungen (einschließlich der bis dato erzielten sozioökonomischen Auswirkungen und weiter gefassten gesellschaftlichen Auswirkungen des Projekts)

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.