Universal graph

Universal graph

In mathematics, a universal graph is an infinite graph that contains "every" finite (or at-most-countable) graph as an induced subgraph. A universal graph of this type was first constructed by R. Rado [cite journal
author = Rado, R.
title = Universal graphs and universal functions
journal = Acta Arithmetica
volume = 9
year = 1964
pages = 331–340] [cite conference
author = Rado, R.
title = Universal graphs
date = 1967
booktitle = A Seminar in Graph Theory
publisher = Holt, Reinhart, and Winston
pages = 83–85] and is now called the Rado graph or random graph. More recent work [cite journal
author = Goldstern, Martin; Kojman, Menachem
title = Universal arrow-free graphs
year = 1996
journal = Acta Math. Hungar.
volume = 1973
pages = 319–326
id = arxiv|archive = math.LO|id = 9409206
doi = 10.1007/BF00052907] [cite journal
author = Cherlin, Greg; Shelah, Saharon; Shi, Niandong
title = Universal graphs with forbidden subgraphs and algebraic closure
journal = Advances in Applied Math
volume = 22
year = 1999
pages = 454–491
id = arxiv|archive = math.LO|id = 9809202
doi = 10.1006/aama.1998.0641] has focused on universal graphs for a graph family "F": that is, an infinite graph belonging to "F" that contains all finite graphs in "F".

A universal graph for a family "F" of graphs can also refer to a member of a sequence of finite graphs that contains all graphs in "F"; for instance, every finite tree is a subgraph of a sufficiently large hypercube graph [cite journal
author = Wu, A. Y.
title = Embedding of tree networks into hypercubes
journal = Journal of Parallel and Distributed Computing
volume = 2
year = 1985
pages = 238–249
doi = 10.1016/0743-7315(85)90026-7] so a hypercube can be said to be a universal graph for trees.

In older mathematical terminology, the phrase "universal graph" was sometimes used to denote a complete graph.

References


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