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+n2 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+n2 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 54 -approximation algorithm for the graphic TSP on simple cubic graphs, improving on the previously best known approximation ratio of 97 . Supported by an NSF Graduate Research Fellowship under Grant No. DGE-1650044 Partially supported by the Natural Sciences and Engineering Research Council of Canada (NSERC), PGSD2532637-2019 Partially supported by NSF Grant DMS 1954134 1 ar X iv :2 11 2. 06 27 8v 1 [ m at h. C O ] 1 2 D ec 2 02 1

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