Thomason's algorithm for finding a second hamiltonian circuit through a given edge in a cubic graph is exponential on Krawczyk's graphs

Abstract

A corollary of Smith's Theorem says that if a cubic graph has a hamiltonian circuit through edge e, then there is a second hamiltonian circuit through e. Tutte's proof is beautiful but non-constructive. Thomason gave a simple and elegant algorithm to find the second hamiltonian circuit. Krawczyk found a family of graphs on which Thomason's algorithm is exponential. I will give a proof that Thomason's algorithm is exponential on a family of cubic planar graphs, which is a variant of Krawczyk's family.