Approximating TSP walks in subcubic graphs

We prove that every simple 2-connected subcubic graph on n vertices with n2 vertices of degree 2 has a TSP walk of length at most 5n+n24−1, confirming a conjecture of Dvořák, Král', and Mohar. This bound is best possible; there are infinitely many subcubic and cubic graphs whose minimum TSP walks have lengths 5n+n24−1 and 5n4−2 respectively. We characterize the extremal subcubic examples meeting this bound. We also give a quadratic-time combinatorial algorithm for finding such a TSP walk. In particular, we obtain a 54-approximation algorithm for the graphic TSP on simple cubic graphs, improving on the previously best known approximation ratio of 97.