Discussiones Mathematicae Graph Theory 33(1) (2013) 33-47
doi: 10.7151/dmgt.1671

Dedicated to Mieczysł aw Borowiecki on his 70th birthday

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Universality for and in Induced-hereditary Graph Properties

Izak Broere

Department of Mathematics and Applied Mathematics
University of Pretoria

Johannes Heidema

Department of Mathematical Sciences
University of South Africa

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 20 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(20) properties in the lattice IK of induced-hereditary properties of which only at most 20 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.

Keywords: countable graph, universal graph, induced-hereditary property

2010 Mathematics Subject Classification: 05C63.

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Received 24 October 2012
Accepted 2 January 2013