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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 chapter focuses on the structure, as opposed to just the existence, of the family of homomorphisms among a set of graphs. The difference is noticeable with even just one graph. For instance, a graph having only the identity homomorphisms to itself is called rigid; rigid graphs are the building blocks of many constructions. Many useful constructions of rigid graphs are provided, and it is shown that asymptotically almost all graphs are rigid; infinite rigid graphs with arbitrary cardinality are also constructed. The homomorphisms among a set of graphs impose the algebraic structure of a category, and every finite category is represented by a set of graphs. This is the generalization of the theorem of Frucht from Chapter 1. Also, as in the case studied by Frucht, it is shown that the representing graphs can be required to have any of a number of graph theoretic properties. However, these properties cannot include having bounded degrees — somewhat surprisingly, since Frucht proved that cubic graphs represent all finite groups.

Keywords:   homomorphism composition, rigid graph, infinite graphs, category, graphs, endomorphism monoid, automorphism group, Isbell’s Condition

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