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On a graph packing conjecture by Bollobás, Eldridge and Catlin

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Abstract

Two graphs G 1 and G 2 of order n pack if there exist injective mappings of their vertex sets into [n], such that the images of the edge sets are disjoint. In 1978, Bollobás and Eldridge, and independently Catlin, conjectured that if (Δ(G 1) + 1)(Δ(G 2) + 1) ≤ n + 1, then G 1 and G 2 pack. Towards this conjecture, we show that for Δ(G 1),Δ(G 2) ≥ 300, if (Δ(G 1) + 1)(Δ(G 2) + 1) ≤ 0.6n + 1, then G 1 and G 2 pack. This is also an improvement, for large maximum degrees, over the classical result by Sauer and Spencer that G 1 and G 2 pack if Δ(G 1)Δ(G 2) < 0.5n.

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References

  1. M. Aigner and S. Brandt: Embedding arbitrary graphs of maximum degree two, J. London Math. Soc. (2) 28 (1993), 39–51.

    Article  MathSciNet  Google Scholar 

  2. N. Alon and E. Fischer: 2-factors in dense graphs, Disc. Math. 152 (1996), 13–23.

    Article  MATH  MathSciNet  Google Scholar 

  3. B. Bollobás: Extremal Graph Theory, Academic Press, London-New York, (1978).

    MATH  Google Scholar 

  4. B. Bollobás and S. E. Eldridge: Packing of graphs and applications to computational complexity, J. Comb. Theory Ser. B 25 (1978), 105–124.

    Article  MATH  Google Scholar 

  5. B. Bollobás, A. Kostochka and K. Nakprasit: Packing d-degenerate graphs, J. Comb. Theory Ser. B 98(1) (2008), 85–94.

    Article  MATH  Google Scholar 

  6. P. A. Catlin: Subgraphs of graphs I., Disc. Math. 10 (1974), 225–233.

    Article  MATH  MathSciNet  Google Scholar 

  7. P. A. Catlin: Embedding subgraphs and coloring graphs under extremal degree conditions, Ph.D. Thesis, Ohio State Univ., Columbus, (1976).

    Google Scholar 

  8. B. Csaba, A. Shokoufandeh and E. Szemerédi: Proof of a conjecture of Bollobás and Eldridge for graphs of maximum degree three, Combinatorica 23(1) (2003), 35–72.

    Article  MATH  MathSciNet  Google Scholar 

  9. N. Eaton: A near packing of two graphs, J. Comb. Theory Ser. B 80 (2000), 98–103.

    Article  MATH  MathSciNet  Google Scholar 

  10. A. Hajnal and E. Szemerédi: Proof of conjecture of Erdős, in: Combinatorial Theory and its Applications, Vol. II (P. Erdős, A. Rényi and V. T. Sós editors), North-Holland, (1970), 601–623.

  11. T. R. Jensen and B. Toft: Graph coloring problems, Wiley-Interscience, (1995).

  12. H. Kaul and A. Kostochka: Extremal graphs for a graph packing theorem of Sauer and Spencer, Combin. Probab. Comput. 16 (2007), 409–416.

    Article  MATH  MathSciNet  Google Scholar 

  13. N. Sauer and J. Spencer: Edge disjoint placement of graphs, J. Combin. Theory Ser. B 25 (1978), 295–302.

    Article  MATH  MathSciNet  Google Scholar 

  14. M. Wozniak: Packing of graphs, Dissertationes Math. 362 (1997), 78 pp.

  15. H. P. Yap: Packing of graphs — a survey, Disc. Math. 72 (1988), 395–404.

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Applied Mathematics, Illinois Institute of Technology, 10 West 32nd Street, Chicago, IL, 60616, USA

    Hemanshu Kaul

  2. Department of Mathematics, University of Illinois, 1409 West Green Street, Urbana, IL, 61801, USA

    Alexandr Kostochka

  3. Institute of Mathematics, Novosibirsk, Russia

    Alexandr Kostochka

  4. Department of Mathematics, Vanderbilt University, 1326 Stevenson Center, Nashville, TN, 37240, USA

    Gexin Yu

Authors
  1. Hemanshu Kaul
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  2. Alexandr Kostochka
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  3. Gexin Yu
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Corresponding author

Correspondence to Alexandr Kostochka.

Additional information

This work was supported in part by NSF grant DMS-0400498. The work of the second author was also partly supported by NSF grant DMS-0650784 and grant 05-01-00816 of the Russian Foundation for Basic Research. The work of the third author was supported in part by NSF grant DMS-0652306.

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Kaul, H., Kostochka, A. & Yu, G. On a graph packing conjecture by Bollobás, Eldridge and Catlin. Combinatorica 28, 469–485 (2008). https://doi.org/10.1007/s00493-008-2278-0

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  • DOI: https://doi.org/10.1007/s00493-008-2278-0

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