Equality of clique-number and chromatic-number extremal functions

Prove that f(δ) = f′(δ) for every integer δ ≥ 4, where f(δ) is the smallest rational constant satisfying diam(G) ≤ f(δ)n + O(1) for all graphs of order n, minimum degree at least δ, and clique number at most 3, and f′(δ) is the corresponding constant for 3-colourable graphs.

Background

The paper defines f(δ) for graphs with clique number at most three and f′(δ) for graphs with chromatic number at most three. Since every graph with clique number at most three need not be 3-colourable, one always has f(δ) ≥ f′(δ).

For δ = 4, 5, and 6, the paper determines the two functions and observes that the optimal repeatable graphs have equal clique and chromatic number. This leads the authors to conjecture equality for all δ ≥ 4.