  We present a new approach for gluing tours over certain tight, 3-edge cuts.
Gluing over 3-edge cuts has been used in algorithms for finding Hamilton cycles
in special graph classes and in proving bounds for 2-edge-connected subgraph
problem, but not much was known in this direction for gluing connected
multigraphs. We apply this approach to the traveling salesman problem (TSP) in
the case when the objective function of the subtour elimination relaxation is
minimized by a $\theta$-cyclic point: $x_e \in \{0,\theta, 1-\theta, 1\}$,
where the support graph is subcubic and each vertex is incident to at least one
edge with $x$-value 1. Such points are sufficient to resolve TSP in general.
For these points, we construct a convex combination of tours in which we can
reduce the usage of edges with $x$-value 1 from the $\frac{3}{2}$ of
Christofides algorithm to $\frac{3}{2}-\frac{\theta}{10}$ while keeping the
usage of edges with fractional $x$-value the same as Christofides algorithm. A
direct consequence of this result is for the Uniform Cover Problem for TSP: In
the case when the objective function of the subtour elimination relaxation is
minimized by a $\frac{2}{3}$-uniform point: $x_e \in \{0, \frac{2}{3}\}$, we
give a $\frac{17}{12}$-approximation algorithm for TSP. For such points, this
lands us halfway between the approximation ratios of $\frac{3}{2}$ of
Christofides algorithm and $\frac{4}{3}$ implied by the famous "four-thirds
conjecture".
