Simple cubic graphs with no short traveling salesman tour · arXivDesk
Abstract
Let t s p ( G ) tsp(G) t s p ( G ) denote the length of a shortest travelling salesman tour in a graph G G G . We prove that for any ε > 0 \varepsilon>0 ε > 0 , there exists a simple 2 2 2 -connected planar cubic graph G 1 G_1 G 1
such that
t s p ( G 1 ) ≥ ( 1.25 − ε ) ⋅ ∣ V ( G 1 ) ∣ tsp(G_1)\ge (1.25-\varepsilon)\cdot|V(G_1)| t s p ( G 1 ) ≥ ( 1.25 − ε ) ⋅ ∣ V ( G 1 ) ∣ , a simple
-connected bipartite cubic graph
such that
t s p ( G 2 ) ≥ ( 1.2 − ε ) ⋅ ∣ V ( G 2 ) ∣ tsp(G_2)\ge (1.2-\varepsilon)\cdot|V(G_2)| t s p ( G 2 ) ≥ ( 1.2 − ε ) ⋅ ∣ V ( G 2 ) ∣ , and a simple
-connected cubic graph
such that
t s p ( G 3 ) ≥ ( 1.125 − ε ) ⋅ ∣ V ( G 3 ) ∣ tsp(G_3)\ge (1.125-\varepsilon)\cdot|V(G_3)| t s p ( G 3 ) ≥ ( 1.125 − ε ) ⋅ ∣ V ( G 3 ) ∣ .