2-walks in 3-connected Planar Graphs

In this we prove that every 3-connected planar graph has closed walk each vertex, none more than twice, such that any vertex visited twice is in a vertex cut of size 3. This both Tutte's Theorem that 4-connected planar graphs are Hamiltonian and the result of Gao and Richter that 3-connected planar graphs have a closed walk visiting each vertex at least once but at most twice.