While the importance of the non-bipartite matching problem itself from an algorithmic and complexity point of view is well known, applications of non-bipartite matching are hard to find.

I did an online search for hints, but almost always the articles I found lacked problems that demonstrated the need for non-bipartite matching. The few exceptions I found were:

A scheduling problem [Fujii, Kasami & Ninomiya, 1969] described in Gerard's matching survey

The oil well drilling problem of [Devine 1973] described in Gerard's matching survey

The Christofides heuristic for the TSP [Christofides 1976] Wikipedia

Plotting street maps with minimum pen lifting [Iri&Taguchi 1980] described in Gerard's matching survey

Kekule structures in chemistry described here

Nonparametric Tests for Homogeneity [Rosenbaum 2005] described here (Wayback Machine)

**TSP: unraveling of tours**with a high number of pairs of crossing edges: Generate a graph $\Gamma$ in which the edges correspond to pairs of crossing tour edges and their weight equals the amount by which the tour length decreases when replacing the respective tour edges by a pair of non-crossing ones.

The maximum weight matching in $\Gamma$ identifies the pairs of tour-edges whose exchanging incurs the maximal length reduction. That matching may have to be applied several times until all crossings have been removed.**Optimizing Triangulations:**take as nodes the triangles and as edges the sides that are shared by two triangles; in that settig a plethora of cost functions can be envisaged for different optimality criteria. There are two basic operations that can be performed on triangulations, namely- diagonal-swapping for the purpose of improving a triangulation:
- minimum weight matching with length reduction incurred by swapping diagonals as edge cost when striving for minimum weight triangulations
- maximum weight matching with the increment of the geometric angle between edge-adjacent triangles when striving for "smooth" triangulations like e.g. of terrain data
- maximum weight matching with the most acute geometric angle of a triangle as edge cost when striving for triangulations of e.g. terrain data, that are in the spirit of planar Delaunay triangulations.

- merging edge-adjacent triangles for the purpose of generating a quadrilateralization:
- a minimum weight matching with the circumference of the generated quadrilateral as edge-cost when striving for "fine grained" quadrilateralizations
- a maximum weight matching over the most acute geometric angle of a generated quadrilateral when striving for the most orthogonal quadrilateralization e.g. for use as finite elements or, as patches for spline interpolation of 3D data.
- a minimum weight matching with the spatial distance between a quadrilateral's diagonals when striving for quadrilateralizations with planar cells.

- diagonal-swapping for the purpose of improving a triangulation:

**Question**

are there further applications for non-bipartite matching, especially newer, examples of different areas of applicability?