A note on induced cycles in Kneser graphs

Combinatorica volume 11, pages245–251(1991)Cite this article

Abstract

Letg(n,r) be the maximal order of an induced cycle in the Knesser graph Kn([n] r), whose vertices are ther-sets of [n]={1, ...,n} and whose adjacency relation is disjointness. Thusg(n, r) is the largestm for which there is a sequenceA 1,A 2,...,A m υ [n] ofr-sets withA i ∩A j=ϑ if and only if |i-j|=1 orm−1. We prove that there is an absolute constantc>0 for which

$$g(2r + 1,r) > c(2.587)^r $$

improving previous results. Our lower bound also shows that the clique covering number of the complement of ann-cycle is at most 1.459 log2 n for large enoughn. Related problems concerning the order of induced subgraphs of bounded degree of Kneser graphs are discussed.

This is a preview of subscription content, access via your institution.

Access options

Buy single article

Instant access to the full article PDF.

US$ 39.95

Tax calculation will be finalised during checkout.

References

  1. [1]

    P. Alles, S. Poljak: Long induced paths and cycles in Kneser graphs, to appear.

  2. [2]

    N. Alon: An extremal problem for a sets with applications to graph theory,J. Comb. Theory (A) 40 (1985), 82–89.

    Google Scholar 

  3. [3]

    N. Alon: Covering graphs by the minimum number of equivalence relations,Combinatorica 6 (1986), 201–206.

    Google Scholar 

  4. [4]

    I. Bárány: A short proof of Knesser's conjecture,J. Combinatorial Theory (A) 25 (1978), 325–326.

    Google Scholar 

  5. [5]

    B. Bollobás: On comnplete subgraphs of different orders,Math. Pro. Cambridge Soc. 79 (1976), 19–24.

    Google Scholar 

  6. [6]

    B. Bollobás:Extremal Graph Theory, Academic Press, London 1978, xx+ 488pp.

    Google Scholar 

  7. [7]

    D. de Caen, D. A. Greagory, N. J. Pullman: Clique coverings of complements of paths and cycles,Annals of Discrete Mathematics 27 (1985), 257–268.

    Google Scholar 

  8. [8]

    P. Erdős, W. A. Goodman, L. Pósa: The representation of a graph by set intersections,Can. J. Math. 18 (1966), 106–112.

    Google Scholar 

  9. [9]

    P. Frankl: An extremal problem for two families of sets,Europ. J. Comb. 3 (1982), 125–127.

    Google Scholar 

  10. [10]

    P. Frankl, andZ. Füredi: Extremal problems concerning Kneser graphs,J. Combinatorial Theory (B) 40 (1986), 270–284.

    Google Scholar 

  11. [11]

    Z. Füredi: Matchings and covers in hypergraphs,Graphs and Combinatorics 4 (1988), 115–206.

    Google Scholar 

  12. [12]

    D. A. Gregory, N. J. Pullman: On a clique covering problem of Orlin,Discrete Math. 41 (1982), 97–99.

    Google Scholar 

  13. [13]

    G. Kalai: Intersection patterns of convex sets,Israel J. Math. 48 (1984), 161–174.

    Google Scholar 

  14. [14]

    M. Kneser: Aufgabe 360,Jahresbericht Deutschen Math. Ver. 58 (1955), 453.

    Google Scholar 

  15. [15]

    L. Lovász: Kneser's conjecture, chromatic number and homotopy,J. Combinatorial Theory (A) 25 (1978), 319–324.

    Google Scholar 

  16. [16]

    S. Poljak, andZs. Tuza: Maximum bipartite subgraphs of Kneser graphs,Graphs and Combinatorics 3 (1987), 191–199.

    Google Scholar 

  17. [16]

    S. Poljak, andZs. Tuza: Maximum bipartite subgraphs of Knesser graphs,Graphs and Combinatorics 3 (1987), 191–199.

    Google Scholar 

Download references

Author information

Affiliations

  1. Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, 16 Mill Lane, CB2 1SB, Cambridge, England

    Y. Kohayakawa

Authors
  1. Y. Kohayakawa
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Research supported by FAPESP, Brazil, Proc. MAP 86/0904-8

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Kohayakawa, Y. A note on induced cycles in Kneser graphs. Combinatorica 11, 245–251 (1991). https://doi.org/10.1007/BF01205076

Download citation

AMS subject classification (1980)

  • 05 C 35
  • 05 C 65