I was reading this blog post about minimum-weight matchings on two-color point sets in the unit square and it got me thinking.
Suppose you have 3 colors (Red, Blue, Green), and randomly drop $N$ points of each color into the unit square. You then draw the minimum-weight matchings for each pair of colors. How many components do you expect the resulting graph to have? What is the average size of a component? Can you say anything else about the distribution of sizes/number of components (aside from the fact that all component sizes are divisible by 3)?
Geometrically: Can two components intersect? Can a component self-intersect? If the answer to both of these questions is No, what can we say about the areas of the regions surrounded by a component?
Addendum: The answer to the latter two questions is Yes, components can intersect and self-intersect. Is there a useful notion of area of a component? What is the expected number of intersections/self-intersections?
Well, I'm going to post some experimental results here, if no one else is going to chime in. This is based off of code I've uploaded here if anyone is interested.
Experimentally (after running a number of tests with $N$ between 100 and 1000), you find that 25-30% of the points wind up in cycles of length 3, with another 8-12% in cycles of length 6, and 5% in cycles of length 9. The average cycle size in this range is somewhere around 9, but there is some indication that it increases with $N$ (and likewise the proportion of points in 3-cycles decreases with $N$). It's a little too early to say for certain though.
The number of intersections and component self-intersections, unsurprisingly, seems to scale with $N^2$, although I can't say much more than that.