Packings in Dense Regular Graphs

  • Authors:

  • Daniela Kühn;Deryk Osthus

  • Affiliations:

  • School of Mathematics and Statistics, Birmingham University, Edgbaston, Birmingham B15 2TT, UK (e-mail: kuehn@maths.bham.ac.uk, osthus@maths.bham.ac.uk);School of Mathematics and Statistics, Birmingham University, Edgbaston, Birmingham B15 2TT, UK (e-mail: kuehn@maths.bham.ac.uk, osthus@maths.bham.ac.uk)

  • Venue:
  • Combinatorics, Probability and Computing
  • Year:
  • 2005

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Abstract

We prove that for all $\alpha,c0$ and for all bipartite graphs $H$, all but at most $\alpha n$ vertices of every $cn$-regular graph $G$ whose order $n$ is sufficiently large can be covered by vertex-disjoint copies of $H$. If the vertex classes of $H$ have different size, then even all but a constant number of vertices of $G$ can be covered. This implies that for all $c0$ and all $r\geq 4$ there exists a constant $C$ such that, in every $cn$-regular graph $G$, all but at most $C$ vertices can be covered by vertex-disjoint subdivisions of $K_r$. We also show that for $r=4,5$ one can take $C=0$.