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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.
Show moreThis 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.
Show more1. 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.
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.
“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)
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