(Georgia Institute of Technology, 2023-07-24)
Wigal, Michael Carroll
A Tutte path of a graph G is a path P of G such that every component of G − P has
at most three attachments on P. Tutte paths are well studied in the literature due to their
applications towards the Hamiltonian cycle problem. We prove the existence of Tutte paths
in which the number of components is bounded for circuit graphs, a natural family of planar
graphs which generalizes 3-connected planar graphs. As a consequence, we obtain a sharp
lower bound for the circumference of essentially 4-connected planar graphs, answering a
conjecture of Fabrici, Harant, Mohr, and Schmidt.
The Traveling Salesperson Problem (TSP) is a foundational problem in the optimization
literature and generalizes the Hamiltonian cycle problem. Motivated by the TSP, we inves-
tigate even covers of subcubic graphs, i.e., finding a small number of cycles that cover the
majority of the vertices (a graph is subcubic if its maximum degree is 3). As an application,
we will show that if G is a 2-connected subcubic graph with n vertices and n_2 vertices of
degree 2, then G has a TSP walk of length at most (5n+n_2)/4−1, establishing a conjecture
of Dvořák, Král', and Mohar. There are an infinite family of subcubic (respectively, cubic)
graphs whose minimum TSP walk have length (5n + n_2)/4 − 1 (respectively, 5n/4 − 2). As
this walk can be found in quadratic time, this provides a state-of-the-art 5/4-approximation
algorithm for the TSP on 2-connected cubic graphs, improving the prior best guarantee of
9/7.