On a graph packing conjecture by Bollobás, Eldridge and Catlin

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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Correspondence to Alexandr Kostochka.

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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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Mathematics Subject Classification (2000)

  • 05C35
  • 05C70