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Perfect packings with complete graphs minus an edge

Oliver Cooley·Daniela Kühn·Deryk Osthus
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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\ge 4 there exists an integer n_0=n_0(r) such that every graph G whose order n\ge n_0 is divisible by r and whose minimum degree is at least (1-1/chi_{cr}(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 chi_{cr}(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.

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Oliver Cooley, Daniela Kühn, Deryk Osthus. 2006-05-08. Perfect packings with complete graphs minus an edge. https://arxiv.org/abs/math/0605189

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