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Graphs and Homomorphisms$
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Pavol Hell and Jaroslav Nesetril

Print publication date: 2004

Print ISBN-13: 9780198528173

Published to Oxford Scholarship Online: September 2007

DOI: 10.1093/acprof:oso/9780198528173.001.0001

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Graphs and Homomorphisms

Pavol Hell

Jaroslav Nešetřil

Oxford University Press

This introductory chapter is a sampler of the material covered in the book. It introduces the notation and terminology in the book, and provides motivational examples and applications, many taken up in more detail in later chapters. It gives the flavour of combinatorial aspects, algorithmic aspects, retractions, duality, constraint satisfaction problems, as well as structural properties of homomorphism composition. The highlights of this chapter include a simple proof of the Colouring Interpolation Theorem, a generalization of the No-Homomorphism Lemma, the construction of a triangle-free graph to which all cubic triangle-free graphs are homomorphic, a case of the Edge Reconstruction Conjecture, and a generalization of a theorem of Frucht on graphs with prescribed automorphism groups.

Keywords:   interpolation, No-Homomorphism Lemma, reconstruction, retraction, core, constraint satisfaction, counting homomorphisms, homomorphism composition, relational systems, automorphism group

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