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{\bf Felix Lazebnik and Spencer Tofts}
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{\bf An Extremal Property of Tur\'an Graphs}
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Let ${\cal F}_{n,t_r(n)}$ denote the family  of all graphs on $n$
vertices and $t_r(n)$ edges,  where $t_r(n)$ is the number of edges
in the Tur\'an's graph $T_r(n)$ -- the complete $r$-partite graph on
$n$ vertices with partition sizes  as equal as possible. For a
graph $G$ and a positive integer $\lambda$, let
$P_G(\lambda)$ denote the number of proper vertex colorings of $G$
with at most $\lambda$ colors, and let $f(n,t_r(n),\lambda) =
\max\{P_G(\lambda):G \in {\cal F}_{n,t_r(n)}\}$. We prove that for all
$n\ge r\ge 2$,  $f(n,t_r(n),r+1) = P_{T_r(n)}(r+1)$ and that
$T_r(n)$ is the only extremal graph.


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