Universality in Graph Properties with Degree Restrictions

• Main Authors: Broere Izak, Heidema Johannes, Mihók Peter
• Format: Article
• Language: English
• Published: Sciendo 2013-07-01
• Series: Discussiones Mathematicae Graph Theory
• Subjects: countable graph; universal graph; induced-hereditary; k-degenerate graph; graph with colouring number at most k + 1; graph property with assignment
• Online Access: https://doi.org/10.7151/dmgt.1696

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.