Sharp results for the Erdős, Pach, Pollack and Tuza problem

Abstract

We consider the Erdős, Pach, Pollack and Tuza problem, asking for the maximum diameter of a graph with given order $n$, minimum degree $δ$ and clique number at most $ω$. We solve their problem asymptotically for the first hard case, $ω\leq 3$, for the smallest values of $δ$ by determining the smallest rational number $f(δ)$ such that $diam(G) \leq f(δ)n+O(1)$ for all graphs $G$ with order $n$, minimum degree $δ$ and clique number $ω\leq 3$. We also consider the weaker version where the clique number $ω\leq 3$ is replaced by having chromatic number $χ\leq 3$ and solve this version for small $δ$, thereby yielding a counterexample to a conjecture of Erdős et al. in a regime where this conjecture was still open. When restricting the conjecture to graphs with chromatic number $χ\leq 3$, we show that this counterexample appears for the smallest possible $δ$, namely $δ=16.$