# Universality for and in Induced-Hereditary Graph Properties

• Volume: 33, Issue: 1, page 33-47
• ISSN: 2083-5892

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## Abstract

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The well-known Rado graph R is universal in the set of all countable graphs I, since every countable graph is an induced subgraph of R. We study universality in I and, using R, show the existence of 2 א0 pairwise non-isomorphic graphs which are universal in I and denumerably many other universal graphs in I with prescribed attributes. Then we contrast universality for and universality in induced-hereditary properties of graphs and show that the overwhelming majority of induced-hereditary properties contain no universal graphs. This is made precise by showing that there are 2(2א0 ) properties in the lattice K ≤ of induced-hereditary properties of which only at most 2א0 contain universal graphs. In a final section we discuss the outlook on future work; in particular the question of characterizing those induced-hereditary properties for which there is a universal graph in the property.

## How to cite

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Izak Broere, and Johannes Heidema. "Universality for and in Induced-Hereditary Graph Properties." Discussiones Mathematicae Graph Theory 33.1 (2013): 33-47. <http://eudml.org/doc/267920>.

@article{IzakBroere2013,
abstract = {The well-known Rado graph R is universal in the set of all countable graphs I, since every countable graph is an induced subgraph of R. We study universality in I and, using R, show the existence of 2 א0 pairwise non-isomorphic graphs which are universal in I and denumerably many other universal graphs in I with prescribed attributes. Then we contrast universality for and universality in induced-hereditary properties of graphs and show that the overwhelming majority of induced-hereditary properties contain no universal graphs. This is made precise by showing that there are 2(2א0 ) properties in the lattice K ≤ of induced-hereditary properties of which only at most 2א0 contain universal graphs. In a final section we discuss the outlook on future work; in particular the question of characterizing those induced-hereditary properties for which there is a universal graph in the property.},
author = {Izak Broere, Johannes Heidema},
journal = {Discussiones Mathematicae Graph Theory},
keywords = {countable graph; universal graph; induced-hereditary property},
language = {eng},
number = {1},
pages = {33-47},
title = {Universality for and in Induced-Hereditary Graph Properties},
url = {http://eudml.org/doc/267920},
volume = {33},
year = {2013},
}

TY - JOUR
AU - Izak Broere
AU - Johannes Heidema
TI - Universality for and in Induced-Hereditary Graph Properties
JO - Discussiones Mathematicae Graph Theory
PY - 2013
VL - 33
IS - 1
SP - 33
EP - 47
AB - The well-known Rado graph R is universal in the set of all countable graphs I, since every countable graph is an induced subgraph of R. We study universality in I and, using R, show the existence of 2 א0 pairwise non-isomorphic graphs which are universal in I and denumerably many other universal graphs in I with prescribed attributes. Then we contrast universality for and universality in induced-hereditary properties of graphs and show that the overwhelming majority of induced-hereditary properties contain no universal graphs. This is made precise by showing that there are 2(2א0 ) properties in the lattice K ≤ of induced-hereditary properties of which only at most 2א0 contain universal graphs. In a final section we discuss the outlook on future work; in particular the question of characterizing those induced-hereditary properties for which there is a universal graph in the property.
LA - eng
KW - countable graph; universal graph; induced-hereditary property
UR - http://eudml.org/doc/267920
ER -

## References

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