Skip to main content

On Six Problems Posed by Jarik Nešetřil

  • Conference paper
Topics in Discrete Mathematics

Part of the book series: Algorithms and Combinatorics ((AC,volume 26))

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. J. Nešetřil, A congruence theorem for asymmetric trees, Pacific Journal of Mathematics 37, 771–778.

    Google Scholar 

  2. J. Nešetřil, Asymmetric graphs and colouring problems, Problems presented at the 6th midsummer combinatorial workshop, Prague 1999.

    Google Scholar 

  3. J. Nešetřil and G. Sabidussi, Minimal asymmetric graphs of induced length 4, Graphs and Combinatorics 8 (1992), 343–359.

    Article  MathSciNet  MATH  Google Scholar 

  4. G. Sabidussi, Clumps, minimal asymmetric graphs, and involutions, J. Gombin. Th. Ser. B 53 (1991), 40–79.

    Article  MATH  MathSciNet  Google Scholar 

  5. P. Wójcik, On automorphisms of digraphs without symmetric cycles, Comment. Math. Univ. Carolinae 37 (1996), 457–467.

    MATH  Google Scholar 

References

  1. L. D. Andersen, The strong chromatic index of a cubic graph is at most 10, Discrete Mathematics 108 (1992), 231–252.

    Article  MATH  MathSciNet  Google Scholar 

  2. R. A. Brualdi and J. J. Quinn Massey, Incidence and strong edge colourings of graphs, Discrete Mathematics 122 (1993), 51–58.

    Article  MATH  MathSciNet  Google Scholar 

  3. F.R. K. Chung, A. Gyárfás, W. T. Trotter, and Z. Tuza, The maximum number of edges in 2K 2-free graphs of bounded degree, Discrete Mathematics 81(1990), 129–135.

    Article  MATH  MathSciNet  Google Scholar 

  4. D. Cranston, A strong edge coloring of graphs of maximum degree 4 using 22 colours, submitted (see www.math.uiuc.edu/~cranston).

    Google Scholar 

  5. P. Erdős and J. Nešetřil, Problem, pp. 162–163 in G. Halász and V. T. Sós (eds.), Irregularities of Partitions, 1989.

    Google Scholar 

  6. R. J. Faudree, A. Gyárfás, R. H. Schelp, and Z. Tuza, Induced Matchings in Bipartite Graphs, Discrete Mathematics 78 (1989), 83–87.

    Article  MathSciNet  MATH  Google Scholar 

  7. R. J. Faudree, A. Gyárfás, R. H. Schelp, and Z. Tuza, The strong chromatic index of graphs, Ars Combinatoria 29B (1990), 205–211.

    Google Scholar 

  8. P. Horak, The strong chromatic index of graphs of maximum degree four, pp. 399–403 in R. Bodendeik (Ed.) Contemporary Methods in Graph Theory, 1990.

    Google Scholar 

  9. P. Horak, H. Qing, and W. T. Trotter, Induced matchings in cubic graphs, Journal of Graph Theory 17 (1993), 151–160.

    MATH  MathSciNet  Google Scholar 

  10. M. Mahdian, The strong chromatic index of C 4-free graphs, Random Structures and Algorithms 17, pp. 357–375, 2000.

    Article  MATH  MathSciNet  Google Scholar 

  11. M. Molloy and B. Reed, A bound on the strong chromatic index of a graph, Journal of Combinatorial Theory (B) 69 (1997), 103–109.

    Article  MATH  MathSciNet  Google Scholar 

References

  1. D. Davenport, N. Hindman, and D. Strauss, Triangle free sets and arithmetic progressions — two Pisier type problems, Electronic J. Combinatorics 9 (2002), no. #R22, 1–19.

    MathSciNet  Google Scholar 

  2. P. Erdős and P. Turán, On some sequences of integers, J. London. Math. Soc. 11 (1936), 261–264.

    Article  Google Scholar 

  3. P. Erdős, J. Nešetřil, and V. Rödl, On colorings and independent sets (Pisier type theorems), submitted.

    Google Scholar 

  4. P. Erdős, J. Nešetřil, and V. Rödl, On Pisier type problems and results (combinatorial applications to number theory), Mathematics of Ramsey theory, Algorithms Combin., vol. 5, Springer, Berlin, 1990, pp. 214–231.

    Google Scholar 

  5. H. Furstenberg, Ergodic behavior of diagonal measures and a theorem of Szemerédi on arithmetic progressions, J. Analyse Math. 31 (1977), 204–256.

    Article  MATH  MathSciNet  Google Scholar 

  6. W. T. Gowers, Hypergraph regularity and the multidimensional Szemerédi theorem, submitted.

    Google Scholar 

  7. W. T. Gowers, A new proof of Szemerédi’s theorem, Geom. Funct. Anal. 11 (2001), no. 3, 465–588.

    MATH  MathSciNet  Google Scholar 

  8. B. Nagle, V. Rödl, and M. Schacht, The counting lemma for regular k-uniform hypergraphs, Random Structures Algorithms 28 (2006), no. 2, 113–179.

    Article  MATH  MathSciNet  Google Scholar 

  9. G. Pisier, Arithmetic characterizations of Sidon sets, Bull. Amer. Math. Soc. (N.S.) 8 (1983), no. 1, 87–89.

    MATH  MathSciNet  Google Scholar 

  10. V. Rödl and J. Skokan, Regularity lemma for k-uniform hypergraphs, Random Structures Algorithms 25 (2004), no. 1, 1–42.

    Article  MATH  MathSciNet  Google Scholar 

  11. E. Szemerédi, On sets of integers containing no k elements in arithmetic progression, Acta Arith. 27 (1975), 199–245, Collection of articles in memory of JuriĬ Vladimirovič Linnik.

    MATH  MathSciNet  Google Scholar 

  12. T. Tao, The dichotomy between structure and randomness, arithmetic progressions, and the primes, submitted.

    Google Scholar 

  13. T. Tao, A quantitative ergodic theory proof of Szemerédi’s theorem, submitted.

    Google Scholar 

  14. B. L. van der Waerden, Beweis einer Baudetschen Vermutung, Nieuw Arch. Wish. 15 (1927), 212–216, German.

    Google Scholar 

References

  1. M. DeVos, J. Nešetřil, and A. Raspaud, On flow and tension-continuous maps, KAM-DIMATIA Series 567 (2002).

    Google Scholar 

  2. M. DeVos and R. Šámal, High girth cubic graphs map to the Clebsch graph, submitted.

    Google Scholar 

  3. A. Galluccio, P. Hell, and J. Nešetřil, The complexity of H-colouring of bounded degree graphs, Discrete Math. 222 (2000), no. 1–3, 101–109.

    Article  MATH  MathSciNet  Google Scholar 

  4. R. Häggkvist and P. Hell, Universality of A-mote graphs, European J. Combin. 14 (1993), no. 1, 23–27.

    Article  MATH  MathSciNet  Google Scholar 

  5. H. Hatami, Random cubic graphs are not homomorphic to the cycle of size 7, J. Combin. Theory Ser. B 93 (2005), no. 2, 319–325.

    Article  MATH  MathSciNet  Google Scholar 

  6. A. V. Kostochka, J. Nešetřil, and P. Smolíková, Colorings and homomorphisms of degenerate and bounded degree graphs, Discrete Math. 233 (2001), no. 1–3, 257–276, Fifth Czech-Slovak International Symposium on Combinatorics, Graph Theory, Algorithms and Applications, (Prague, 1998).

    Article  MATH  MathSciNet  Google Scholar 

  7. J. Nešetřil, Aspects of structural combinatorics (graph homomorphisms and their use), Taiwanese J. Math. 3 (1999), no. 4, 381–423.

    MathSciNet  MATH  Google Scholar 

  8. I. M. Wanless and N. C. Wormald, Regular graphs with no homomorphisms onto cycles, J. Combin. Theory Ser. B 82 (2001), no. 1, 155–160.

    Article  MATH  MathSciNet  Google Scholar 

References

  1. N. Alon, M. Krivelevich and P. Seymour, Long cycles in critical graphs, J. Graph Theory 35 (2000), 193–196.

    Article  MATH  MathSciNet  Google Scholar 

  2. M. Chudnovsky, N. Robertson, P. Seymour and R. Thomas, The strong perfect graph theorem, Annals of Mathematics, to appear.

    Google Scholar 

  3. G. A. Dirac, The structure of k-chromatic graphs, Fund. Math. 40 (1953), 42–55.

    MATH  MathSciNet  Google Scholar 

  4. G.A. Dirac, Circuits in critical graphs, Monatsh. Math. 59 (1955), 178–187.

    Article  MATH  MathSciNet  Google Scholar 

  5. G. A. Dirac, On the structure of 5-and 6-chromatic abstract graphs, J. Reine Angew. Math. 214/215 (1964), 43–52.

    MathSciNet  Google Scholar 

  6. T. Gallai, Kritische Graphen I, Publ. Math. Inst. Hungar. Acad. Sci. 8 (1963), 165–192.

    MATH  MathSciNet  Google Scholar 

  7. G. Hajós, Über eine Konstruktion nicht n-färbbarer Graphen, Wiss. Z. Martin-Luther-Univ. Halle-Wittenberg Math.-Natur. Reihe 10 (1961), 116–117.

    Google Scholar 

  8. J. B. Kelly and L. M. Kelly, Paths and circuits in critical graphs, Amer. J. Math. 76 (1954), 786–792.

    Article  MATH  MathSciNet  Google Scholar 

  9. R. C. Reid, Maximal circuits in critical graphs, J. London Math. Soc. 32 (1957), 456–462.

    Article  MathSciNet  Google Scholar 

  10. M. Stiebitz, Subgraphs of colour-critical graphs, Gombinatorica 7 (1987), 303–312.

    Article  MATH  MathSciNet  Google Scholar 

  11. B. Toft, Some contributions to the theory of colour-critical graphs, Ph.D.-thesis University of London 1970, published by Aarhus University in Various Publication Series 14 (1970).

    Google Scholar 

  12. B. Toft, On critical subgraphs of colour-critical graphs, Discrete Math. 7 (1974), 377–392.

    MATH  MathSciNet  Google Scholar 

  13. B. Toft, Some problems and results related to subgraphs of colour critical graphs, in: R. Bodendiek, H. Schumacher and G. Walther (editors), Graphen in Forschung und Unterricht, Festschrift K. Wagner, Barbara Franzbecker Verlag 1985, 178–186.

    Google Scholar 

  14. H.-J. Voss, Graphs with prescribed maximal subgraphs and critical chromatic graphs, Comment. Math. Univ. Carolinae 18 (1977), 129–142.

    MATH  MathSciNet  Google Scholar 

  15. H.-J. Voss, Cycles and bridges in graphs, Deutscher Verlag der Wissenschaften, Kluwer Academic Publishers, 1991.

    Google Scholar 

  16. W. Wessel, Some colour-critical equivalents of the strong perfect graph conjecture, in: Proc. Int. Koll. Graphentheorie und deren Anwendungen, Oberhof DDR, Matematisches Gesellschaft der DDR 1977, 300–309.

    Google Scholar 

  17. W. Wessel, Critical lines, critical graphs and odd cycles, Technical report, Akademie der Wissenschaften der DDR, Institut für Mathematik 1981.

    Google Scholar 

References

  1. K. I. Aardal, S. P. M. van Hoesel, A. M. C. A. Kosterand, C. Mannino, and A. Sassano, Models and solution techniques for frequency assignment problems, Technical report, Konrad-Zuse-Zentrum für Informationstech-nik Berlin, 2001.

    Google Scholar 

  2. P. K. Agarwal and N. H. Mustafa, Independent set of intersection graphs of convex objects in 2d, in 9th Scandinavian Workshop on Algorithm Theory (SWAT’04), pages 127–137, 2004.

    Google Scholar 

  3. P. K. Agarwal, M. van Kreveld, and S. Suri, Label Placement by Maximum Independent Set in Rectangles, pages 233–238, 1997.

    Google Scholar 

  4. C. Ambühl and U. Wagner, The clique problem in intersection graphs of ellipses and triangles, Theory Comput. Syst, 38(3):279–292, 2005.

    Article  MATH  MathSciNet  Google Scholar 

  5. H. Breu and D. G. Kirkpatrick, Unit disk graph recognition is NP-hard. Computational Geometry. Theory and Applications, 9(1–2):3–24, 1998.

    MATH  MathSciNet  Google Scholar 

  6. T. M. Chan, Polynomial-time approximation schemes for packing and piercing fat objects, J. Algorithms, 46(2):178–189, 2003.

    Article  MATH  MathSciNet  Google Scholar 

  7. B. N. Clark, C. J. Colbourn, and D. S. Johnson, Unit disk graphs, Discrete Mathematics, 86:165–177, 1990.

    Article  MATH  MathSciNet  Google Scholar 

  8. T. Erlebach, K. Jansen, and E. Seidel, Polynomial-time approximation schemes for geometric intersection graphs, SIAM J. Comput., 34(6):1302–1323, 2005.

    Article  MATH  MathSciNet  Google Scholar 

  9. W. K. Hale, Frequency assignment: Theory and applications, Proceedings of the IEEE, 68(12):1497–1514, December 1980.

    Article  Google Scholar 

  10. J. Håstad, Clique is hard to approximate within n 1-ε, Aca Math., 182(1):105–142, 1999.

    Article  MATH  Google Scholar 

  11. P. Hliněný and J. Kratochvíl, Representing graphs by disks and balls (a survey of recognition-complexity results), Discrete Math., 229(1–3):101–124, 2001.

    Article  MathSciNet  MATH  Google Scholar 

  12. H.B. Hunt, III, M.V. Marathe, V. Radhakrishnan, S.S. Ravi, D.J. Rosenkrantz, and R. E. Stearns, NC-approximation schemes for NP-and PSPACE-hard problems for geometric graphs, J. Algorithms, 26(2):238–274, 1998.

    Article  MATH  MathSciNet  Google Scholar 

  13. J. Kratochvíl and A. Kuběna, On intersection representations of co-planar graphs, Discrete Math., 178(1–3):251–255, 1998.

    Article  MATH  MathSciNet  Google Scholar 

  14. J. Kratochvíl and J. Matoušek, Intersection graphs of segments, J. Combin. Theory Ser. B, 62(2):289–315, 1994.

    Article  MATH  MathSciNet  Google Scholar 

  15. J. Kratochvíl and J. Nešetřil, Independent set and clique problems in intersection-defined classes of graphs, Comment. Math. Univ. Carolin., 31(1):85–93, 1990.

    MATH  MathSciNet  Google Scholar 

  16. E. Malesińska, Graph-Theoretical Models for Frequency Assignment Problems, PhD thesis, TU Berlin, 1997.

    Google Scholar 

  17. M. Middendorf and F. Pfeiffer, The max clique problem in classes of string-graphs, Discrete Math., 108(1–3):365–372, 1992.

    Article  MATH  MathSciNet  Google Scholar 

  18. T. Nieberg, J. Hurink, and W. Kern, A robust PTAS for maximum weight independent sets in unit disk graphs, In Proceedings of the 30th International Workshop on Graph-Theoretic Concepts in Computer Science (WG’04), 2004.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2006 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Bang-Jensen, J., Reed, B., Schacht, M., Šámal, R., Toft, B., Wagner, U. (2006). On Six Problems Posed by Jarik Nešetřil. In: Klazar, M., Kratochvíl, J., Loebl, M., Matoušek, J., Valtr, P., Thomas, R. (eds) Topics in Discrete Mathematics. Algorithms and Combinatorics, vol 26. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-33700-8_30

Download citation

Publish with us

Policies and ethics