DIMACS TR: 2001-51

Proof of a Tiling Conjecture of Komlos

Authors: Ali Shokoufandeh and Yi Zhao


Given two graphs $G$ and $H$, an $H$-{\em matching} of $G$ (or a {\em tiling} of $G$ with $H$) is a subgraph of $G$ consisting of vertex-disjoint copies of $H$. For an $r$-chromatic graph $H$ on $h$ vertices, we write $u=u(H)$ for the smallest possible color-class size in any $r$-coloring of $H$. The {\em critical chromatic number} of $H$ is the number $\chi_{cr}(H)=(r-1)h/(h-u)$. A conjecture of Koml\'{o}s states that for every graph $H$, there is a constant $K$ such that if $G$ is any $n$-vertex graph of minimum degree at least $\left(1-(1/\chi_{cr}(H))\right)n$, then $G$ contains an $H$-matching that covers all but $K$ vertices of $G$. In this paper we prove that the conjecture holds for all sufficiently large values of $n$.

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