On the maximum diameter of $k$-colorable graphs

TL;DR

This paper investigates the maximum diameter of graphs with certain forbidden subgraphs and colorability constraints, providing counterexamples to existing conjectures and establishing new upper bounds for $k$-colorable graphs.

Contribution

It constructs counterexamples to a conjecture on diameters of $K_{2r}$-free graphs and proves new diameter bounds for $k$-colorable graphs with minimum degree constraints.

Findings

Counterexamples to the Erdős-Pach-Pollack-Tuza conjecture for certain parameters.

Upper bounds on the diameter of $k$-colorable graphs with minimum degree.

Specific diameter bounds for 3-colorable graphs.

Abstract

Erd\H{o}s, Pach, Pollack and Tuza [J. Combin. Theory, B 47, (1989), 279-285] conjectured that the diameter of a $K_{2r}$-free connected graph of order $n$ and minimum degree $\delta\geq 2$ is at most $\frac{2(r-1)(3r+2)}{(2r^2-1)}\cdot \frac{n}{\delta} + O(1)$ for every $r\ge 2$, if $\delta$ is a multiple of $(r-1)(3r+2)$. For every $r>1$ and $\delta\ge 2(r-1)$, we create $K_{2r}$-free graphs with minimum degree $\delta$ and diameter $\frac{(6r-5)n}{(2r-1)\delta+2r-3}+O(1)$, which are counterexamples to the conjecture for every $r>1$ and $\delta>2(r-1)(3r+2)(2r-3)$. The rest of the paper proves positive results under a stronger hypothesis, $k$-colorability, instead of being $K_{k+1}$-free. We show that the diameter of connected $k$-colorable graphs with minimum degree $\geq \delta$ and order $n$ is at most $\left(3-\frac{1}{k-1}\right)\frac{n}{\delta}+O(1)$, while for $k=3$, it is at most $\frac{57n}{23\delta}+O\left(1\right)$.