Budden M. Star-Critical Ramsey Numbers for Graphs 2023

Textbook in PDF format

This text is a comprehensive survey of the literature surrounding star-critical Ramsey numbers. First defined by Jonelle Hook in her 2010 dissertation, these numbers aim to measure the sharpness of the corresponding Ramsey numbers by determining the minimum number of edges needed to be added to a critical graph for the Ramsey property to hold. Despite being in its infancy, the topic has gained significant attention among Ramsey theorists.
This work provides researchers and students with a resource for studying known results and their complete proofs. It covers typical results, including multicolor star-critical Ramsey numbers for complete graphs, trees, cycles, wheels, and n-good graphs, among others. The proofs are streamlined and, in some cases, simplified, with a few new results included. The book also explores the connection between star-critical Ramsey numbers and deleted edge numbers, which focus on destroying the Ramsey property by removing edges.
The book concludes with open problems and conjectures for researchers to consider, making it a valuable resource for those studying the field of star-critical Ramsey numbers.
Preface
List of Figures
Introduction to Multicolor Star-Critical Ramsey Numbers
Graph-Theoretic Background
Adding or Deleting Edges
Ramsey-Goodness
Star-Critical Ramsey Numbers Involving Various Graphs
Trees Versus Trees
Paths Versus Paths
Stars Versus Stars
Other Trees
Trees Versus Non-Trees
Trees Versus Complete Graphs
Paths Versus Non-Trees
Cycles
Cycles Versus Cycles
Cycles Versus Other Graphs
Wheels
Disjoint Unions of Complete Graphs
Fans
Books
Generalized Fans and Books
Generalizations of Star-Critical Ramsey Numbers
Star-Critical Gallai-Ramsey Numbers
Deleting Subgraphs Other Than Stars
Directions for Future Research and Summary of Known Values
Directions for Future Study
Summary of Bounds and Tables of Known Values
References
Index