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 I of all countable graphs (since every graph in I is isomorphic to an induced subgraph of R).
In this paper we describe a general recursive construction which proves the existence of a countable universal graph for any induced-hereditary property of countable general graphs. A general construction of a universal graph for the set of finite graphs in a product of properties of graphs is also presented.
The paper is concluded by a comparison between the two types of construction of universal graphs.
[1] ALLEN, P.— LOZIN, V.— RAO, M.: Clique-width and the speed of hereditary properties, Electron. J. Combin. 16 (2009), 1–11. 10.37236/124Search in Google Scholar
[2] BOROWIECKI, M.— BROERE, I.— FRICK, M.— SEMANIŠIN, G.— MIHÓK, P.: A survey of hereditary properties of graphs, Discuss. Math. Graph Theory 17 (1997), 5–50. http://dx.doi.org/10.7151/dmgt.103710.7151/dmgt.1037Search in Google Scholar
[3] BROERE, I.— HEIDEMA, J.: Constructing an abundance of Rado graphs, Util. Math. 84 (2011), 139–152. Search in Google Scholar
[4] BROERE, I.— HEIDEMA, J.: Some universal directed labelled graphs, Util. Math. 84 (2011), 311–324. Search in Google Scholar
[5] BROERE, I.— HEIDEMA, J.— MIHÓK, P.: Universality in graph properties with degree restrictions, Discuss. Math. Graph Theory (To appear). Search in Google Scholar
[6] CAMERON, P. J.: The random graph revisited. In: Australian Mathematical Society Winter Meeting in Brisbane, and European Congress of Mathematics in Barcelona, July 2000. http://www.maths.qmw.ac.uk/~pjc/slides/rgr.pdf 10.1007/978-3-0348-8268-2_15Search in Google Scholar
[7] CHERLIN, G.— KOMJÁTH, P.: There is no universal countable pentagon-free graph, J. Graph Theory 18 (1994) 337–341. http://dx.doi.org/10.1002/jgt.319018040510.1002/jgt.3190180405Search in Google Scholar
[8] DIESTEL, R.: Graph Theory (4th ed.), Springer, Heidelberg, 2010. http://dx.doi.org/10.1007/978-3-642-14279-610.1007/978-3-642-14279-6Search in Google Scholar
[9] HAJNAL, A.— PACH, J.: Monochromatic paths in infinite coloured graphs. In: Finite and Infinite Sets, Eger (Hungary). Colloq. Math. Soc. János Bolyai 37, (1981), 359–369. 10.1016/B978-0-444-86893-0.50028-0Search in Google Scholar
[10] KOMJÁTH, P.— PACH, J.: Universal graphs without large bipartite subgraphs, Mathematika 31 (1984) 282–290. http://dx.doi.org/10.1112/S002557930001250X10.1112/S002557930001250XSearch in Google Scholar
[11] LOZIN, V.— RUDOLF, G.: Minimal universal bipartite graphs, Ars Combin. 84 (2007), 345–356. Search in Google Scholar
[12] MIHÓK, P.— MIŠKUF, J.— SEMANIŠIN, G.: On universal graphs for hom-properties, Discuss. Math. Graph Theory 29 (2009), 401–409. http://dx.doi.org/10.7151/dmgt.145510.7151/dmgt.1455Search in Google Scholar
[13] RADO, R.: Universal graphs and universal functions, Acta Arith. 9 (1964), 331–340. 10.4064/aa-9-4-331-340Search in Google Scholar
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