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

TL;DR

The paper resolves the first hard case of the Erdős–Pach–Pollack–Tuza diameter problem for graphs with clique number at most $3$ by introducing repeatable graphs, fundamental blocks, and concatenation to characterize the asymptotic diameter ratio $f(\delta)$. It first derives exact bounds for $\omega\le 3$ in the $K_4$-free regime, obtaining $f(4)=\frac{4}{7}$, $f(5)=\frac{5}{11}$, and $f(6)=\frac{14}{37}$ through a detailed neighbourhood and block analysis, including tightness via block concatenations. For the weaker $\chi\le 3$ variant, the authors determine $f'(7)=\frac{17}{52}$ and $f'(8)=\frac{2}{7}$ and provide a lower bound $f'(16)\ge \frac{31}{216}$, yielding a counterexample to Erdős et al.'s conjecture in this regime, with comparisons to the known CSS21 bound $f'(\delta)\le \frac{7}{3\delta}$. The work combines rigorous combinatorial arguments with computer-assisted searches to identify optimal repeatable blocks, advancing understanding of how degree, clique, and colorability constraints shape diameter growth in large graphs.

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 $δ=16.$

Figures (4)

• Figure 1: The graph $G$ (where vertices in $N_0$ are shown larger) and its concatenation $G'$
• Figure 2: Repetitive concatenation of the repeatable graph (on the left)
• Figure 3: $K_4$-free graphs with $\delta=4$ and large diameter for small orders used in \ref{['prop:omega3Delta4exact']}
• Figure 4: The optimal graph for $\delta=5$. The red line draws attention to the missing edge.

Theorems & Definitions (35)

• Theorem 1
• proof : Proof of \ref{['thr:EPPT_main']} for $\omega \leq 2$
• Conjecture 2: EPPT89
• Conjecture 3: CSS21
• Conjecture 4
• Proposition 5
• proof
• Claim 6
• proof : Proof
• Proposition 7