Transversals in Linear Uniform Hypergraphs

ISBN-13: 9783030465612 / Angielski / Miękka / 2021 / 229 str.

Michael A. Henning; Anders Yeo

Opis

This book gives the state-of-the-art on transversals in linear uniform hypergraphs. The notion of transversal is fundamental to hypergraph theory and has been studied extensively. Very few articles have discussed bounds on the transversal number for linear hypergraphs, even though these bounds are integral components in many applications. This book is one of the first to give strong non-trivial bounds on the transversal number for linear hypergraphs.  The discussion may lead to further study of those problems which have not been solved completely, and may also inspire the readers to raise new questions and research directions.

The book is written with two readerships in mind. The first is the graduate student who may wish to work on open problems in the area or is interested in exploring the field of transversals in hypergraphs. This exposition will go far to familiarize the student with the subject, the research techniques, and the major accomplishments in the field.  The photographs included allow the reader to associate faces with several researchers who made important discoveries and contributions to the subject. The second audience is the established researcher in hypergraph theory who will benefit from having easy access to known results and latest developments in the field of transversals in linear hypergraphs.

Recenzje

“The book is well written, all the proofs are clear and often illustrated with pictures, that makes them easier to follow. I recommend this book to the researchers in the area of hypergraphs – it may serve as a nice survey on the recent results about the transversals in linear hypergraphs. I also think that it would be very useful for graduate students that would like to get familiar with this topic.” (Marcin Anholcer, zbMATH 1454.05003, 2021)

Spis treści

1. Introduction.- 2. Linear Intersection Hypergraphs.- 3. Finite Affine Planes and Projective Planes.- 4 . The Tuza Constants.- 5. The Tuza Constant c4.- 6. The Tuza Constant ck for k Large.- 7. The West Bound.- 8. The Deficiency of a Hypergraph.- 9. The Tuza Constant q4.- 10. The Tuza Constant qk for Large k.- 11. The Cap Set Problem.- 12. Partial Steiner Triple Systems.- 13. Upper Transversals in Linear Hypergraphs.- 14. Strong Tranversals in Linear Hypergraphs.- 15. Conjectures and Open Problems.- References.- Glossary.

O autorze

Michael A. Henning is a world-leader in domination theory in graphs. He has been a plenary and invited speaker at several international conferences and is a prolific researcher having published over 460 papers to date in international mathematics journals. Michael was born and schooled in South Africa having obtained his PhD at the University of Natal in April 1989. In January 1989, he started his academic career as a lecturer at the University of Zululand, before accepting a lectureship in mathematics at the former University of Natal in January 1991. In January 2000 Michael was appointed a Full Professor at the University of Natal, which later merged with the University of Durban-Westville to form the University of KwaZulu-Natal in January 2004. After spending almost 20 years at the University of KwaZulu-Natal and one of its predecessors, the University of Natal, Michael moved to the University of Johannesburg in May 2010 as a research professor.

​Anders Yeo is a world-leader in several areas within mathematics and computer science, including total domination in graphs and transversal in hypergraphs, digraphs and algorithms. He has been a plenary and invited speaker at several international conferences and is a prolific researcher having published over 150 papers to date in international mathematics journals. Anders was born in Australia and schooled in Denmark having obtained his PhD at Odense University in December 1997.  Thereafter Anders has worked at the University of Victoria in Canada, the University of Aarhus in Denmark, Royal Holloway, University of London, in the UK, the University of Johannesburg in South Africa and the Singapore University of Technology and Design in Singapore.  Currently Anders is a professor of Mathematics at the University of Southern Denmark in Denmark.