Discussiones Mathematicae Graph Theory 33(3) (2013) 477-492
doi: 10.7151/dmgt.1696

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Universality in graph properties with degree restrictions

Izak Broere

Department of Mathematics and Applied Mathematics
University of Pretoria
Pretoria, South Africa

Johannes Heidema

Department of Mathematical Sciences
University of South Africa
Pretoria, South Africa

Peter Mihók

Department of Applied Mathematics
Technical University of Košice
Košice, Slovakia


Rado constructed a (simple) denumerable graph R with the positive integers as vertex set with the following edges: For given m and n with m < n, m is adjacent to n if n has a 1 in the m'th position of its binary expansion. It is well known that R is a universal graph in the set ℑc of all countable graphs (since every graph in ℑc is isomorphic to an induced subgraph of R).

A brief overview of known universality results for some induced-hereditary subsets of ℑc is provided. We then construct a k-degenerate graph which is universal for the induced-hereditary property of finite k-degenerate graphs. In order to attempt the corresponding problem for the property of countable graphs with colouring number at most k+1, the notion of a property with assignment is introduced and studied. Using this notion, we are able to construct a universal graph in this graph property and investigate its attributes.

Keywords: countable graph, universal graph, induced-hereditary, k-degenerate graph, graph with colouring number at most k+1, graph property with assignment

2010 Mathematics Subject Classification: 05C63.


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Received 28 February 2012
Revised 29 October 2012
Accepted 29 October 2012