Erdős et al. (1989) [4] conjectured that the diameter of a $K_{2r}$-free connected graph of order n and minimum degree $\delta \ge 2$ is at most $\frac{2(r-1)(3r+2)}{(2r^2-1)}\cdot \frac{n}{\delta }+O(1)$ for every $r\ge 2$, if δ 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 δ and diameter $\frac{(6r-5)n}{(2r-1)\delta +2r-3}+O(1)$, which are counterexamples to the conjecture for every $r>1$ and $\delta >2(r-1)(3r+2)(2r-3)$.

中文翻译：

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Erdős、Pach、Pollack 和 Tuza 猜想的反例

Erdős 等人。(1989) [4] 推测 a 的直径${钾}_{2r}$-n阶和最小度的自由连通图$\delta \ge 2$ 最多是 $\frac{2(r-1)(3r+2)}{(2r^2-1)}\cdot \frac{n}{\delta }+哦(1)$ 对于每个 $r\ge 2$, 如果δ是$(r-1)(3r+2)$. 对于每$r>1$ 和 $\delta \ge 2(r-1)$，我们创建 ${钾}_{2r}$具有最小度数δ和直径的自由图$\frac{(6r-5)n}{(2r-1)\delta +2r-3}+哦(1)$，这是每个猜想的反例 $r>1$ 和 $\delta >2(r-1)(3r+2)(2r-3)$.