Simple cubic graphs with no short traveling salesman tour

TL;DR

This paper constructs specific classes of simple cubic graphs with high ratios of shortest TSP tours to the number of vertices, demonstrating limitations in TSP tour efficiency for these graph types.

Contribution

It introduces explicit constructions of simple cubic graphs with no short TSP tours, establishing lower bounds on TSP tour lengths relative to graph size.

Findings

Existence of 2-connected planar cubic graphs with TSP length at least 1.25 times the number of vertices.

Existence of bipartite cubic graphs with TSP length at least 1.2 times the number of vertices.

Existence of 3-connected cubic graphs with TSP length at least 1.125 times the number of vertices.

Abstract

Let $tsp(G)$ denote the length of a shortest travelling salesman tour in a graph $G$. We prove that for any $\varepsilon>0$, there exists a simple $2$-connected planar cubic graph $G_1$ such that $tsp(G_1)\ge (1.25-\varepsilon)\cdot|V(G_1)|$, a simple $2$-connected bipartite cubic graph $G_2$ such that $tsp(G_2)\ge (1.2-\varepsilon)\cdot|V(G_2)|$, and a simple $3$-connected cubic graph $G_3$ such that $tsp(G_3)\ge (1.125-\varepsilon)\cdot|V(G_3)|$.