Approximating TSP walks in subcubic graphs

12/12/2021

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by   Michael C. Wigal, et al.

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We prove that every simple 2-connected subcubic graph on n vertices with n_2 vertices of degree 2 has a TSP walk of length at most 5n+n_2/4-1, 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+n_2/4-1 and 5n/4 - 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 5/4-approximation algorithm for the graphic TSP on simple cubic graphs, improving on the previously best known approximation ratio of 9/7.