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

Abstract

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