Generalized diameter bound for bounded clique or chromatic number

Establish whether every connected graph of order n, minimum degree at least δ, and clique number at most k satisfies diam(G) ≤ (3 − 2/k)n/δ + O(1) for every k ≥ 3 and δ ≥ 3k/2 − 1, and likewise determine whether the same bound holds under the weaker assumption that the chromatic number is at most k.

Background

Conjecture 3 is presented as an updated version of the earlier Erdős–Pach–Pollack–Tuza conjecture. It proposes a uniform asymptotic upper bound on the diameter of connected graphs with prescribed minimum degree and bounded clique number, together with a weaker variant for bounded chromatic number.

The paper notes that the weaker chromatic-number version had already been proved for k equal to 3 and 4, while the general statement remains the broader conjectural framework motivating the investigation of small clique and chromatic numbers.