The minimum degree threshold for perfect graph packings

Abstract

Let H be any graph. We determine up to an additive constant the minimum degree of a graph G which ensures that G has a perfect H-packing (also called an H-factor). More precisely, let δ(H,n) denote the smallest integer k such that every graph G whose order n is divisible by |H| and with δ(G)≥k contains a perfect H-packing. We show that

$$ \delta (H,n) = \left( {1 - \frac{1} {{\chi ^ * (H)}}} \right)n + O(1) $$

.

The value of χ*(H) depends on the relative sizes of the colour classes in the optimal colourings of H and satisfies χ(H)−1<χ*(H)≤χ(H).

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Correspondence to Daniela Kühn.

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Kühn, D., Osthus, D. The minimum degree threshold for perfect graph packings. Combinatorica 29, 65–107 (2009). https://doi.org/10.1007/s00493-009-2254-3

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Mathematics Subject Classification (2000)

  • 05C70
  • 05C35