Counterexamples to two conjectures on the diameter of clique-free graphs

Source abstract

Erdős et al. (JCT-B, 1989) conjectured that, for integers $r\ge 2$ and $δ\ge 2$ with $3r-1\midδ$, every connected $K_{2r+1}$-free graph of order $n$ and minimum degree $δ$ has diameter at most $\frac{3r-1}{r}\cdot \frac{n}δ+O(1)$. Czabarka et al. (JCT-B, 2021) later proposed the following generalization: for every $k\ge 3$ and $δ\ge\left\lceil\frac{3k}{2}\right\rceil-1$, every connected $K_{k+1}$-free graph of order $n$ and minimum degree at least $δ$ has diameter at most $(3-\frac{2}{k})\cdot\frac{n}δ+O(1)$. We disprove the latter conjecture, including its $k$-colorable version, for every $k\ge 7$ and sufficiently large $δ$. When $k=2r\ge 8$ and $3r-1\midδ$, our construction also disproves the conjecture of Erdős et al. (JCT-B, 1989).