Approximating TSP walks in subcubic graphs

TL;DR

This paper proves a tight bound on the length of TSP walks in subcubic graphs, characterizes extremal cases, and provides a quadratic-time algorithm that improves approximation ratios for the graphic TSP.

Contribution

It establishes the exact maximum TSP walk length in simple 2-connected subcubic graphs and offers a new efficient approximation algorithm for the graphic TSP.

Findings

Proves the bound of (5n + n_2)/4 - 1 for TSP walks in subcubic graphs.

Characterizes extremal graphs meeting this bound.

Provides a quadratic-time algorithm for finding such TSP walks.

Abstract

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 $\frac{5n+n_2}{4}-1$, confirming a conjecture of Dvo\v{r}\'ak, Kr\'al', and Mohar. This bound is best possible; there are infinitely many subcubic and cubic graphs whose minimum TSP walks have lengths $\frac{5n+n_2}{4}-1$ and $\frac{5n}{4} - 2$ 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 $\frac{5}{4}$-approximation algorithm for the graphic TSP on simple cubic graphs, improving on the previously best known approximation ratio of $\frac{9}{7}$.