# Ordering tours in a Euclidean TSP according to (strictly) increasing length

Let $$H$$ be the set of all Hamiltonian cycles on the complete graph $$K_n$$ associated with a set of $$n \geq 4$$ points $$P$$ in the plane where edge weights are defined using the Euclidean distance between pairs of points in $$P$$. Suppose that $$h_1 \in H$$ is an optimal tour with length $$d(h_1)$$ and that the tours are numbered in order of increasing length so that: $$d(h_1) \leq d(h_2) \leq \cdots \leq d(h_{|H|})$$.

Question: Is there a set of conditions on the set of points in $$P$$ that would result in the ordering above being a sequence of strict inequalities? I am interested in finding a lower bound $$\lambda < 1$$ that characterizes the ratio $$d(h_1) / d(h_2) < \lambda$$ given some distribution of points, if possible. Intuitively it seems this may be related to the minimum distance between any pair of points in $$P$$, or the minimum difference in tour length resulting from an edge swap. Perhaps there are some other well known results on the distribution of tour lengths that might be related. I would be grateful for any references that come to mind on topics related to this question.

• One of the inequalities is wrongly stated (currently, they are contradictory). – Gamow Nov 27 '18 at 12:38