The minimum degree threshold for perfect graph packings

  • Authors:

  • Daniela Kühn;Deryk Osthus

  • Affiliations:

  • Birmingham University, School of Mathematics, B15 2TT, Edgbaston Birmingham, UK;Birmingham University, School of Mathematics, B15 2TT, Edgbaston Birmingham, UK

  • Venue:
  • Combinatorica
  • Year:
  • 2009

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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)−1χ*(H)≤χ(H).