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Universality in graph properties with degree restrictions. (English) Zbl 1274.05334

Summary: 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 \(\mathcal I_c\) of all countable graphs (since every graph in \(\mathcal I_c\) is isomorphic to an induced subgraph of \(R\)).
A brief overview of known universality results for some induced-hereditary subsets of \(\mathcal I_c\) 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.

MSC:

05C63 Infinite graphs
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