DISTANCE-UNIFORM GRAPHS WITH LARGE DIAMETER.

An ∊-distance-uniform graph is one with a critical distance d such that from every vertex, all but at most an ∊-fraction of the remaining vertices are at distance exactly d. Motivated by the theory of network creation games, Alon, Demaine, Hajiaghayi, and Leighton made the following conjecture of independent interest: that every ∊-distance-uniform graph (and, in fact, a broader class of ∊-distance-almost-uniform graphs) has critical distance at most logarithmic in the number of vertices n. We disprove this conjecture and characterize the asymptotics of this extremal problem. Specifically, for 1/n ≤ ∊ ≤ 1/logn, we construct ∊-distance-uniform graphs with critical distance 2Ω(logn/loge-1) 2O(logn/log e-1). We also prove an upper bound on the critical distance of the form 2Ologn/loge-1 for all e and n. Our lower bound construction introduces a novel method inspired by the Tower of Hanoi puzzle and may itself be of independent interest. [ABSTRACT FROM AUTHOR]