Universality for and in Induced-Hereditary Graph Properties
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 gra...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
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University of Zielona Góra
2013-03-01
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| Series: | Discussiones Mathematicae Graph Theory |
| Subjects: | |
| Online Access: | https://doi.org/10.7151/dmgt.1671 |
| Summary: | 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. |
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| ISSN: | 2083-5892 |