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We consider a problem proposed by Linial and Wilf to determine the structureof graphs that allows the maximum number of $q$-colorings among graphs with $n$vertices and $m$ edges. Let $T_r(n)$ denote the Tur\'{a}n graph - the complete$r$-partite graph on $n$ vertices with partition sizes as equal as possible. Weprove that for all odd integers $q\geq 5$ and sufficiently large $n$, theTur\'{a}n graph $T_2(n)$ has at least as many $q$-colorings as any other graph$G$ with the same number of vertices and edges as $T_2(n)$, with equalityholding if and only if $G=T_2(n)$. Our proof builds on methods by Norine and byLoh, Pikhurko, and Sudakov, which reduces the problem to a quadratic program.

The extremality of 2-partite Tur\'an graphs with respect to the number of colorings