Extension of lower bounds to non-hereditary edge and planar (non-hereditary) discrepancy

Determine whether the Ω(n^{1/4})-type lower bound established for hereditary discrepancy of unique shortest path systems in undirected graphs extends to (non-hereditary) edge discrepancy in undirected graphs and to (non-hereditary) vertex or edge discrepancy in planar graphs.

Background

The authors obtain tight (up to polylogarithmic factors) lower bounds of order n^{1/4} for hereditary discrepancy of unique shortest paths in undirected graphs and strengthen these to non-hereditary vertex discrepancy in undirected and directed settings. They also show hereditary vertex discrepancy lower bounds for planar graphs.

However, whether analogous n^{1/4}-scale lower bounds hold for non-hereditary edge discrepancy in undirected graphs and for non-hereditary vertex or edge discrepancy in planar graphs remains unresolved.