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[1][2] and is now called the Rado graph or random graph. More recent work[3] [4] 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[5] 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
- ^ Rado, R. (1964). "Universal graphs and universal functions". Acta Arithmetica. 9: 331–340.
- ^ Rado, R. (1967). "Universal graphs". A Seminar in Graph Theory. Holt, Reinhart, and Winston. pp. 83–85.
{{cite conference}}: Unknown parameter|booktitle=ignored (|book-title=suggested) (help) - ^ Goldstern, Martin; Kojman, Menachem (1996). "Universal arrow-free graphs". Acta Math. Hungar. 1973: 319–326. doi:10.1007/BF00052907. arXiv:math.LO/9409206.
{{cite journal}}: CS1 maint: multiple names: authors list (link) - ^ Cherlin, Greg; Shelah, Saharon; Shi, Niandong (1999). "Universal graphs with forbidden subgraphs and algebraic closure". Advances in Applied Math. 22: 454–491. doi:10.1006/aama.1998.0641. arXiv:math.LO/9809202.
{{cite journal}}: CS1 maint: multiple names: authors list (link) - ^ Wu, A. Y. (1985). "Embedding of tree networks into hypercubes". Journal of Parallel and Distributed Computing. 2: 238–249. doi:10.1016/0743-7315(85)90026-7.