  Erd\H{o}s, Pach, Pollack and Tuza [J. Combin. Theory, B 47, (1989), 279-285]
conjectured that the diameter of a $K_{2r}$-free connected graph of order $n$
and minimum degree $\delta\geq 2$ is at most
$\frac{2(r-1)(3r+2)}{(2r^2-1)}\cdot \frac{n}{\delta} + O(1)$ for every $r\ge
2$, if $\delta$ is a multiple of $(r-1)(3r+2)$. For every $r>1$ and $\delta\ge
2(r-1)$, we create $K_{2r}$-free graphs with minimum degree $\delta$ and
diameter $\frac{(6r-5)n}{(2r-1)\delta+2r-3}+O(1)$, which are counterexamples to
the conjecture for every $r>1$ and $\delta>2(r-1)(3r+2)(2r-3)$. The rest of the
paper proves positive results under a stronger hypothesis, $k$-colorability,
instead of being $K_{k+1}$-free. We show that the diameter of connected
$k$-colorable graphs with minimum degree $\geq \delta$ and order $n$ is at most
$\left(3-\frac{1}{k-1}\right)\frac{n}{\delta}+O(1)$, while for $k=3$, it is at
most $\frac{57n}{23\delta}+O\left(1\right)$.
