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

Erdős et al. (1989) [4] conjectured that the diameter of a K2r-free connected graph of order n and minimum degree δ≥2 is at most 2(r−1)(3r+2)(2r2−1)⋅nδ+O(1) for every r≥2, if δ is a multiple of (r−1)(3r+2). For every r>1 and δ≥2(r−1), we create K2r-free graphs with minimum degree δ and diameter (6r−5)n(2r−1)δ+2r−3+O(1), which are counterexamples to the conjecture for every r>1 and δ>2(r−1)(3r+2)(2r−3).