Optimal cyclic crossing-number constant

Determine the largest constant c_t in [0,1] such that, for every integer t≥3 and every cyclic sequence of t distinct non-isolated B-side vertices v_1,…,v_t satisfying c_{v_i,v_{i+1}}≤c_{v_{i+1},v_i} for every i (with v_{t+1}=v_1), the maximum of the reverse-to-forward crossing-number ratios around the cycle is at least c_t.

Background

For two B-side vertices u and v, the crossing numbers c_{u,v} and c_{v,u} measure crossings involving edges incident to the two vertices under the respective relative order. Question 2 considers a directed cycle of t≥3 vertices in which each forward crossing number is no larger than the corresponding reverse crossing number.

The desired constant c_t is important because a uniform positive lower bound independent of t would imply that GREEDY is a constant-factor approximation algorithm for OSCM. Theorem 4 establishes the lower bound c_t≥1/(t−1), and Theorem 5 determines the sharp value for t=3 under the additional assumption that the vertices have pairwise disjoint neighborhoods, but the general problem remains unresolved.