Perfect packings with complete graphs minus an edge

Authors:
Oliver Cooley;Daniela Kühn;Deryk Osthus
Affiliations:
School of Mathematics, Birmingham University, Edgbaston, Birmingham B15 2TT, UK;School of Mathematics, Birmingham University, Edgbaston, Birmingham B15 2TT, UK;School of Mathematics, Birmingham University, Edgbaston, Birmingham B15 2TT, UK
Venue:
European Journal of Combinatorics
Year:
2007

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Abstract

Let K"r^- denote the graph obtained from K"r by deleting one edge. We show that for every integer r=4 there exists an integer n"0=n"0(r) such that every graph G whose order n=n"0 is divisible by r and whose minimum degree is at least (1-1/@g"c"r(K"r^-))n contains a perfect K"r^--packing, i.e. a collection of disjoint copies of K"r^- which covers all vertices of G. Here @g"c"r(K"r^-)=r(r-2)r-1 is the critical chromatic number of K"r^-. The bound on the minimum degree is best possible and confirms a conjecture of Kawarabayashi for large n.