Universality in Graph Properties with Degree Restrictions

Abstract

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