Erdős–Pach–Pollack–Tuza diameter conjecture for K-free graphs

Prove or determine the validity of the Erdős–Pach–Pollack–Tuza diameter bounds for connected K_{2r}-free and K_{2r+1}-free graphs of order n and fixed minimum degree δ in the parameter ranges specified by Conjecture 2, including the regimes not excluded by the known counterexamples.

Background

The paper recalls a conjecture of Erdős, Pach, Pollack, and Tuza concerning the largest possible diameter of a connected graph in terms of its order and minimum degree when the graph is restricted by clique number. Part (i) concerns K_{2r}-free graphs and part (ii) concerns K_{2r+1}-free graphs, with asymptotic bounds depending on r and δ.

Previously known counterexamples disprove part (i) for sufficiently large δ, but the cited work leaves an intermediate parameter regime unresolved. The present paper studies the first difficult case, particularly graphs with clique number at most three, and supplies an additional counterexample in a regime where the conjecture had remained unresolved.