Star-critical Ramsey numbers for graphs
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 cr...
| Main Author: | Budden, Mark R. |
|---|---|
| Other Authors: | SpringerLink (Online service) |
| Format: | eBook |
| Language: | English |
| Published: |
Cham :
Springer,
2023.
|
| Physical Description: |
1 online resource (102 pages). |
| Series: |
SpringerBriefs in mathematics.
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| Subjects: |
| Summary: |
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. |
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| Item Description: |
Intro -- Preface -- Contents -- List of Figures -- 1 Introduction to Multicolor Star-Critical Ramsey Numbers -- 1.1 Graph-Theoretic Background -- 1.2 Adding or Deleting Edges -- 1.3 Ramsey-Goodness -- 2 Star-Critical Ramsey Numbers Involving Various Graphs -- 2.1 Trees Versus Trees -- 2.1.1 Paths Versus Paths -- 2.1.2 Stars Versus Stars -- 2.1.3 Other Trees -- 2.2 Trees Versus Non-Trees -- 2.2.1 Trees Versus Complete Graphs -- 2.2.2 Paths Versus Non-Trees -- 2.3 Cycles -- 2.3.1 Cycles Versus Cycles -- 2.3.2 Cycles Versus Other Graphs -- 2.4 Wheels -- 2.5 Disjoint Unions of Complete Graphs. 2.6 Fans -- 2.7 Books -- 2.8 Generalized Fans and Books -- 3 Generalizations of Star-Critical Ramsey Numbers -- 3.1 Star-Critical Gallai-Ramsey Numbers -- 3.2 Deleting Subgraphs Other Than Stars -- 4 Directions for Future Research and Summary of Known Values -- 4.1 Directions for Future Study -- 4.2 Summary of Bounds and Tables of Known Values -- References -- Index. 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. Includes bibliographical references and index. |
| Physical Description: |
1 online resource (102 pages). |
| Bibliography: |
Includes bibliographical references and index. |
| ISBN: |
9783031299810 3031299817 |