DMGT

ISSN 1234-3099 (print version)

ISSN 2083-5892 (electronic version)

https://doi.org/10.7151/dmgt

Discussiones Mathematicae Graph Theory

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Discussiones Mathematicae Graph Theory

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Discussiones Mathematicae Graph Theory 19(2) (1999) 199-217
DOI: 10.7151/dmgt.1095

GENERALIZED RAMSEY THEORY AND DECOMPOSABLE PROPERTIES OF GRAPHS

Stefan A. Burr

Department of Computer Science
City College, C.U.N.Y. New York, NY 10031, U.S.A.

e-mail: burr@cs-mail.engr.ccny.cuny.edu

Michael S. Jacobson

University of Louisville
Louisville, KY 40292, U.S.A.

e-mail: mikej@luisville.edu

Peter Mihók

Mathematical Institute, Slovak Academy of Sciences
Gresákova 6, 040 01 Košice, Slovak Republic
e-mail: mihok@Košice.upjs.sk

Gabriel Semanišin

Department of Geometry and Algebra
Faculty of Science, P.J. Safárik University
Jesenná 5, 041 54 Košice, Slovak Republic
e-mail: semanisin@duro.upjs.sk

Abstract

In this paper we translate Ramsey-type problems into the language of decomposable hereditary properties of graphs. We prove a distributive law for reducible and decomposable properties of graphs. Using it we establish some values of graph theoretical invariants of decomposable properties and show their correspondence to generalized Ramsey numbers.

Keywords: hereditary properties, additivity, reducibility, decomposability, Ramsey number, graph invariants.

1991 Mathematics Subject Classification: 05C15, 05C55, 05C75.

References

[1] M. Borowiecki, I. Broere, M. Frick, P. Mihók and G. Semanišin, Survey of hereditary properties of graphs, Discuss. Math. Graph Theory 17 (1997) 5-50, doi: 10.7151/dmgt.1037.
[2] M. Borowiecki and M. Hałuszczak, Decompositions of some classes of graphs (manuscript) 1998.
[3] S.A. Burr, A ramsey-theoretic result involving chromatic numbers, J. Graph Theory 4 (1980) 241-242, doi: 10.1002/jgt.3190040212.
[4] S.A. Burr and M.S. Jacobson, Arrow relations involving partition parameters of graphs (manuscript) 1982.
[5] S.A. Burr and M.S. Jacobson, On inequalities involving vertex-partition parameters of graphs, Congressus Numerantium 70 (1990) 159-170.
[6] R.L. Graham, M. Grötschel and L. Lovász, Handbook of combinatorics (Elsevier Science B.V., Amsterdam, 1995).
[7] T.R. Jensen and B. Toft, Graph colouring problems (Wiley-Interscience Publications, New York, 1995).
[8] T. Kövari, V.T. Sós and P. Turán, On a problem of K. Zarankiewicz, Colloq. Math. 3 (1969) 50-57.
[9] J. Kratochvíl, P. Mihók and G. Semanišin, Graphs maximal with respect to hom-properties, Discuss. Math. Graph Theory 17 (1997) 77-88, doi: 10.7151/dmgt.1040.
[10] R. Lick and A.T. White, k-degenerate graphs, Canad. J. Math. 22 (1970) 1082-1096; MR42#1715.
[11] L. Lovász, On decomposition of graphs, Studia Sci. Math. Hungar 1 (1966) 237-238; MR34#2442.
[12] P. Mihók, On graphs critical with respect to vertex partition numbers, Discrete Math. 37 (1981) 123-126, doi: 10.1016/0012-365X(81)90146-1.
[13] P. Mihók and G. Semanišin, On the chromatic number of reducible hereditary properties (submitted).
[14] M. Simonovits, Extremal graph theory, in: L.W. Beineke and R.J. Wilson, eds., Selected topics in graph theory vol. 2 (Academic Press, London, 1983) 161-200.

Received 2 February 1999
Revised 8 September 1999