Counterexamples to a conjecture of Erdős, Pach, Pollack and Tuza

Abstract

Erdős et al. (1989) [4] conjectured that the diameter of a $K_{2r}$-free connected graph of order n and minimum degree $\delta \ge 2$ is at most $\frac{2(r-1)(3r+2)}{(2r^2-1)}\cdot \frac{n}{\delta }+O(1)$ for every $r\ge 2$, if δ 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 δ and diameter $\frac{(6r-5)n}{(2r-1)\delta +2r-3}+O(1)$, which are counterexamples to the conjecture for every $r>1$ and $\delta >2(r-1)(3r+2)(2r-3)$.

Introduction

The following theorem was discovered several times [1], [4], [6], [7]:

Theorem 1

For a fixed minimum degree $\delta \ge 2$ and $n\to \infty$, for every n-vertex connected graph G, we have $\mathrm{diam}(G)\le \frac{3n}{\delta +1}+O(1)$.

Note that the upper bound is sharp (even for δ-regular graphs [2]), but the constructions have complete subgraphs whose order increases with δ. Erdős, Pach, Pollack, and Tuza [4] conjectured that the upper bound in Theorem 1 can be strengthened for graphs not containing complete subgraphs:

Conjecture 1 [4]

Let $r,\delta \ge 2$ be fixed integers and let G be a connected graph of order n and minimum degree δ.

• If G is $K_{2r}$-free and δ is a multiple of $(r-1)(3r+2)$ then, as $n\to \infty$,

  $$\mathrm{diam}(G)\le \frac{2(r-1)(3r+2)}{(2r^2-1)}\cdot \frac{n}{\delta }+O(1)=(3-\frac{2}{2r-1}-\frac{1}{(2r-1)(2r^2-1)})\frac{n}{\delta }+O(1).$$

• If G is $K_{2r+1}$-free and δ is a multiple of $3r-1$, then, as $n\to \infty$,

  $$\mathrm{diam}(G)\le \frac{3r-1}{r}\cdot \frac{n}{\delta }+O(1)=(3-\frac{2}{2r})\frac{n}{\delta }+O(1).$$

Set $k=2r$ or $k=2r+1$ according the cases. As connected δ-regular graphs are $K_{\delta +1}$-free (apart from $K_{\delta +1}$ itself), we need $\delta \ge k$ (at least) to make improvement on Theorem 1. Furthermore, as the conjectured constants in the bounds are at most $3-\frac{2}{k}$, Theorem 1 implies that the conjectured inequalities hold trivially, unless $\delta \ge \frac{3k}{2}-1$.

Erdős et al. [4] constructed graph sequences for every $r,\delta \ge 2$, where δ satisfies the divisibility condition, which meet the upper bounds in Conjecture 1. We show these construction them in Section 2.

Part (ii) of Conjecture 1 for $r=1$ was proved in Erdős et al. [4]. Conjecture 1 is included in the book of Fan Chung and Ron Graham [5], which collected Erdős's significant problems in graph theory.

No more progress has been reported on this conjecture, except that for $r=2$ in (ii), under a stronger hypothesis (4-colorable instead of $K_5$-free), Czabarka, Dankelman and Székely [3] arrived at the conclusion of Conjecture 1:

Theorem 2

For every connected 4-colorable graph G of order n and minimum degree $\delta \ge 1$, $\mathrm{diam}(G)\le \frac{5n}{2\delta }-1$.

In Section 3, we give an unexpected counterexample for Conjecture 1 (i) for every $r\ge 2$ and $\delta >2(r-1)(3r+2)(2r-3)$. The question whether Conjecture 1 (i) holds in the range $(r-1)(3r+2)\le \delta \le 2(r-1)(3r+2)(2r-3)$ is still open. The counterexample leads to a modification of Conjecture 1, which no longer requires cases:

Conjecture 2

For every $k\ge 3$ and $\delta \ge \lceil \frac{3k}{2}\rceil -1$, if G is a $K_{k+1}$-free (weaker version: k-colorable) connected graph of order n and minimum degree at least δ, $\mathrm{diam}(G)\le (3-\frac{2}{k})\frac{n}{\delta }+O(1)$.

For $k=2r$, Conjecture 2 is identical to Conjecture 1(ii). For $k=2r-1$, $3-\frac{2}{k}=\frac{6r-5}{2r-1}$, and, although the conjectured bound is likely not tight for any δ, the fraction $\frac{6r-5}{2r-1}$ cannot be reduced for all δ according to the construction in Section 3.