Consider $n$ points arbitrarily located on the plane. Consider a random graph $G$ drawn from $G(n, \frac12)$ on these points (i.e. the Erdos-Renyi random graph where every edge is selected with probability $\frac12$).

What is known about the geometry of the minimum spanning tree of such a graph? I am interested in pointers to any literature on this, but something like the following might be a concrete example:

Thm. With high probability, the Minimum Spanning Tree has weight within a factor of $\alpha$ of the MST on the complete graph on the same points.
  • 2
    $\begingroup$ Related : mathoverflow.net/questions/38824 $\endgroup$ Sep 14, 2012 at 18:13
  • $\begingroup$ My guess is that the size of the minimal spanning tree in your random graph ought to be fairly close (within a multiplicative constant) of the size of the euclidean minimal spanning tree, since only half the edges are missing. So you might start by looking at J. Michael Steele, "Growth Rates of Euclidean Minimal Spanning Trees with Power Weighted Edges". $\endgroup$ Sep 14, 2012 at 21:15
  • $\begingroup$ Thanks for the pointers! They seem to be only slightly different models (randomly placed points etc), but I will definitely take a look. $\endgroup$
    – Pradipta
    Sep 16, 2012 at 3:29


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