Universality in Graph Properties with Degree Restrictions

• Volume: 33, Issue: 3, page 477-492
• ISSN: 2083-5892

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Abstract

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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.

How to cite

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Izak Broere, Johannes Heidema, and Peter Mihók. "Universality in Graph Properties with Degree Restrictions." Discussiones Mathematicae Graph Theory 33.3 (2013): 477-492. <http://eudml.org/doc/268159>.

@article{IzakBroere2013,
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.},
author = {Izak Broere, Johannes Heidema, Peter Mihók},
journal = {Discussiones Mathematicae Graph Theory},
keywords = {countable graph; universal graph; induced-hereditary; k-degenerate graph; graph with colouring number at most k + 1; graph property with assignment; -degenerate graph; graph with colouring number at most },
language = {eng},
number = {3},
pages = {477-492},
title = {Universality in Graph Properties with Degree Restrictions},
url = {http://eudml.org/doc/268159},
volume = {33},
year = {2013},
}

TY - JOUR
AU - Izak Broere
AU - Johannes Heidema
AU - Peter Mihók
TI - Universality in Graph Properties with Degree Restrictions
JO - Discussiones Mathematicae Graph Theory
PY - 2013
VL - 33
IS - 3
SP - 477
EP - 492
AB - 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.
LA - eng
KW - countable graph; universal graph; induced-hereditary; k-degenerate graph; graph with colouring number at most k + 1; graph property with assignment; -degenerate graph; graph with colouring number at most
UR - http://eudml.org/doc/268159
ER -

References

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