# Universality for and in Induced-Hereditary Graph Properties

Discussiones Mathematicae Graph Theory (2013)

- Volume: 33, Issue: 1, page 33-47
- ISSN: 2083-5892

## Access Full Article

top## Abstract

top## How to cite

topIzak Broere, and Johannes Heidema. "Universality for and in Induced-Hereditary Graph Properties." Discussiones Mathematicae Graph Theory 33.1 (2013): 33-47. <http://eudml.org/doc/267920>.

@article{IzakBroere2013,

abstract = {The well-known Rado graph R is universal in the set of all countable graphs I, since every countable graph is an induced subgraph of R. We study universality in I and, using R, show the existence of 2 א0 pairwise non-isomorphic graphs which are universal in I and denumerably many other universal graphs in I with prescribed attributes. Then we contrast universality for and universality in induced-hereditary properties of graphs and show that the overwhelming majority of induced-hereditary properties contain no universal graphs. This is made precise by showing that there are 2(2א0 ) properties in the lattice K ≤ of induced-hereditary properties of which only at most 2א0 contain universal graphs. In a final section we discuss the outlook on future work; in particular the question of characterizing those induced-hereditary properties for which there is a universal graph in the property.},

author = {Izak Broere, Johannes Heidema},

journal = {Discussiones Mathematicae Graph Theory},

keywords = {countable graph; universal graph; induced-hereditary property},

language = {eng},

number = {1},

pages = {33-47},

title = {Universality for and in Induced-Hereditary Graph Properties},

url = {http://eudml.org/doc/267920},

volume = {33},

year = {2013},

}

TY - JOUR

AU - Izak Broere

AU - Johannes Heidema

TI - Universality for and in Induced-Hereditary Graph Properties

JO - Discussiones Mathematicae Graph Theory

PY - 2013

VL - 33

IS - 1

SP - 33

EP - 47

AB - The well-known Rado graph R is universal in the set of all countable graphs I, since every countable graph is an induced subgraph of R. We study universality in I and, using R, show the existence of 2 א0 pairwise non-isomorphic graphs which are universal in I and denumerably many other universal graphs in I with prescribed attributes. Then we contrast universality for and universality in induced-hereditary properties of graphs and show that the overwhelming majority of induced-hereditary properties contain no universal graphs. This is made precise by showing that there are 2(2א0 ) properties in the lattice K ≤ of induced-hereditary properties of which only at most 2א0 contain universal graphs. In a final section we discuss the outlook on future work; in particular the question of characterizing those induced-hereditary properties for which there is a universal graph in the property.

LA - eng

KW - countable graph; universal graph; induced-hereditary property

UR - http://eudml.org/doc/267920

ER -

## References

top- [1] M. Borowiecki, I. Broere, M. Frick, G. Semanišin and P. Mihók, A survey of hereditary properties of graphs, Discuss. Math. Graph Theory 17 (1997) 5-50. doi:10.7151/dmgt.1037[Crossref] Zbl0902.05026
- [2] I. Broere and J. Heidema, Constructing an abundance of Rado graphs, Util. Math. 84 (2011) 139-152. Zbl1228.05220
- [3] I. Broere and J. Heidema, Some universal directed labelled graphs, Util. Math. 84 (2011) 311-324. Zbl1228.05250
- [4] I. Broere and J. Heidema, Universal H-colourable graphs, accepted for publication in Graphs Combin. doi:10.1007/s00373-012-1216-5[Crossref] Zbl1272.05136
- [5] I. Broere and J. Heidema, Induced-hereditary graph properties, homogeneity, extensibility and universality, accepted for publication in J. Combin. Math. Combin. Comput. Zbl1274.05400
- [6] I. Broere, J. Heidema and P. Mihók, Constructing universal graphs for inducedhereditary graph properties, accepted for publication in Math. Slovaca. Zbl1324.05135
- [7] I. Broere, J. Heidema and P. Mihók, Universality in graph properties with degree restrictions, accepted for publication in Discuss. Math. Graph Theory. Zbl1274.05334
- [8] P.J. Cameron, The random graph revisited, http://www.math.uni-bielefeld.de/rehmann/ECM/cdrom/3ecm/pdfs/pant3/camer.pdf Zbl1023.05124
- [9] G. Cherlin and P. Komjáth, There is no universal countable pentagon-free graph, J. Graph Theory 18 (1994) 337-341. doi:10.1002/jgt.3190180405[Crossref] Zbl0805.05044
- [10] G. Cherlin, S. Shelah and N. Shi, Universal graphs with forbidden subgraphs and algebraic closure, Adv. Appl. Math. 22 (1999) 454-491. doi:10.1006/aama.1998.0641[Crossref] Zbl0928.03049
- [11] G. Cherlin and N. Shi, Graphs omitting a finite set of cycles, J. Graph Theory 21 (1996) 351-355. doi:10.1002/(SICI)1097-0118(199603)21:3h351::AID-JGT11i3.0.CO;2-K[Crossref] Zbl0845.05060
- [12] M. Chudnovsky, N. Robertson, P. Seymour and R. Thomas, The strong perfect graph theorem, Ann. Math. 164 (2006) 51-229. doi:10.4007/annals.2006.164.51[Crossref] Zbl1112.05042
- [13] B.A. Davey and H.A. Priestly, Introduction to Lattices and Order, Second Edition, (Cambridge University Press, New York, 2008).
- [14] R. Diestel, Graph Theory, Fourth Edition, Graduate Texts in Mathematics, 173, (Springer, Heidelberg, 2010).
- [15] R. Fraïssé, Sur l’extension aux relations de quelques propriétiés connues des ordres, C. R. Acad. Sci. Paris 237 (1953) 508-510. Zbl0053.02904
- [16] A. Hajnal and J. Pach, Monochromatic paths in infinite coloured graphs, in: Colloquia Mathematica Societatis J´anos Bolyai 37, Finite and infinite sets, Eger (Hungary) (1981), 359-369.
- [17] C.W. Henson, A family of countable homogeneous graphs Pacific J. Math. 38 (1971) 69-83. doi:10.2140/pjm.1971.38.69[Crossref]
- [18] P. Komjáth and J. Pach, Universal graphs without large bipartite subgraphs, Mathematika 31 (1984) 282-290. doi:10.1112/S002557930001250X[Crossref] Zbl0551.05057
- [19] F.R. Madelaine, Universal structures and the logic of forbidden patterns, Log. Methods Comput. Sci. 5 (2:13) (2009) 1-25. doi:10.2168/LMCS-5(2:13)2009[Crossref][WoS]
- [20] P. Mihók, J. Miškuf and G. Semanišin, On universal graphs for hom-properties, Discuss. Math. Graph Theory 29 (2009) 401-409. doi:10.7151/dmgt.1455[Crossref] Zbl1194.05041
- [21] R. Rado, Universal graphs and universal functions, Acta Arith. 9 (1964) 331-340. Zbl0139.17303

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.