Download PDF - On generating sets of induced-hereditary propertiesAll rights reserved. Copyright exceptions and limitations apply

ArticleOriginal scientific text

Title

On generating sets of induced-hereditary properties

Authors 1

Affiliations

  1. Department of Geometry and Algebra, Faculty of Science, P.J. Šafárik University, Jesenná 5, 041 54 Košice, Slovak Republic

Abstract

A natural generalization of the fundamental graph vertex-colouring problem leads to the class of problems known as generalized or improper colourings. These problems can be very well described in the language of reducible (induced) hereditary properties of graphs. It turned out that a very useful tool for the unique determination of these properties are generating sets. In this paper we focus on the structure of specific generating sets which provide the base for the proof of The Unique Factorization Theorem for induced-hereditary properties of graphs.

Keywords

ENG
induced-hereditary property of graphs, additivity, reducibility, generating sets, maximal graphs, unique factorization

Bibliography

  1. B. Bollobás and A.G. Thomason, Hereditary and monotone properties of graphs, in: R.L. Graham and J. Nesetril, eds., The mathematics of Paul Erdős, II, Algorithms and Combinatorics 14 (Springer-Verlag, 1997) 70-78.
  2. 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.
  3. M. Borowiecki and P. Mihók, Hereditary properties of graphs, in: V.R. Kulli, ed., Advances in Graph Theory (Vishwa International Publication, Gulbarga, 1991) 42-69.
  4. I. Broere and J.Bucko, Divisibility in additive hereditary graph properties and uniquely partitionable graphs, Tatra Mountains Math. Publications 18 (1999) 79-87.
  5. G. Chartrand, D. Geller and S. Hedetniemi, Graphs with forbidden subgraphs, J. Combin. Theory (B) 10 (1971) 12-41, doi: 10.1016/0095-8956(71)90065-7.
  6. E.J. Cockayne, Color classes for r-graphs, Canad. Math. Bull. 15 (3) (1972) 349-354, doi: 10.4153/CMB-1972-063-2.
  7. E.J. Cockayne, G.G. Miller and G. Prins, An interpolation theorem for partitions which are complete with respect to hereditary properties, J. Combin. Theory (B) 13 (1972) 290-297, doi: 10.1016/0095-8956(72)90065-2.
  8. M. Frick, A survey of (m,k)-colorings, in: J. Gimbel c.a, ed., Quo Vadis, Graph Theory? A source book for challenges and directions, Annals of Discrete Mathematics 55 (North-Holland, Amsterdam, 1993) 45-57.
  9. S.T. Hedetniemi, On hereditary properties of graphs, J. Combin. Theory (B) 14 (1973) 94-99, doi: 10.1016/S0095-8956(73)80009-7.
  10. T.R. Jensen and B. Toft, Graph colouring problems (Wiley-Interscience Publications, New York, 1995).
  11. P. Mihók, Additive hereditary properties and uniquely partitionable graphs, in: M. Borowiecki and Z. Skupień, eds., Graphs, hypergraphs and matroids (Zielona Góra, 1985) 49-58.
  12. P. Mihók, Unique factorization theorem, Discuss. Math. Graph Theory 20 (2000) 143-154, doi: 10.7151/dmgt.1114.
  13. P. Mihók, G. Semanišin and R. Vasky, Additive and Hereditary Properties of Graphs are Uniquely Factorizable into Irreducible Factors, J. Graph Theory 33 (2000) 44-53, doi: 10.1002/(SICI)1097-0118(200001)33:1<44::AID-JGT5>3.0.CO;2-O
  14. J. Mitchem, Maximal k-degenerate graphs, Utilitas Math. 11 (1977) 101-106.
Discussiones Mathematicae Graph Theory
2002/22/1
Export to PBN, Opens in new tab
Pages:
183-192
Main language of publication
English
Received
2000-07-31
Accepted
2001-05-21
Published
2002

DOI: 10.7151/dmgt.1167

Exact and natural sciences