Schnyder Woods and Long Induced Paths in 3-Connected Planar Graphs

In the recent 30 years, Schnyder woods have become an invaluable asset in the study of planar graphs. We contribute to this research with a brief and comprehensible proof of a new structural feature: every Schnyder wood of a 3-connected planar graph on n vertices has a tree of depth at least \(\lfloor 1/6 \log _2 n\rfloor \) . As a simple implication, our result improves the previous hard-won lower bound on the length of an induced path in such a graph to \(1/6 \log _2 n \) .