Sharp results for the Erd\H{o}s, Pach, Pollack and Tuza problem

TL;DR

This paper asymptotically solves the maximum diameter problem for graphs with small clique number and minimum degree, providing new bounds and counterexamples to a longstanding conjecture.

Contribution

It determines the asymptotic maximum diameter bounds for graphs with clique number at most 3 and small minimum degree, and constructs counterexamples to a conjecture for the first time.

Findings

Asymptotic bounds for diameter with clique number ≤ 3

Counterexample to Erdős et al. conjecture at δ=16

Solution for the weak chromatic number ≤ 3 version

Abstract

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