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Let $K_{(m,n)}$ be the complete bipartite graph with partitions of size $m$ and $n$. How many spanning subgraphs does $K_{(m,n)}$ have?

I found a similar question asked here:

https://mathoverflow.net/questions/98721/number-of-spanning-subgraphs-of-the-complete-bipartite-graph-km-n

and one of the suggestion is to use the Polya Enumeration Theorem or the Burnside Lemma to find the number of spanning subgraphs.

I'm not that familiar with the theorems mentioned so I studied them but I'm still not sure how to apply them to the problem.

My idea is since there are $mn$ edges in $K_{(m,n)}$, I can just color them by white or black. Every coloring of the edges will produce a spanning subgraph. The problem now is that some may be isomorphic to each other. To apply PET I need to have a permutation group. There lies my problem now. What group should I use?

Hoping for some hints or ideas. Thank you!

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chowching
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  • As Ira Gessel points out in the comments to the MO post the problem of bi-coloring the edges of a bipartite graph up to isomorphism (symmetric group acting on the two vertex sets) is not difficult and indeed can be found on pages $94-100$ of Harary and Palmer, Graphical Enumeration. However I am not sure what you mean by saying these colorings correspond to spanning subgraphs. Say in a bipartite graph you color all the edges incident on one vertex white and the rest black, then neither the white edges taken together nor the black ones span the graph. – Marko Riedel Feb 22 '15 at 20:46
  • Thanks. I have to check if we have that book in the library. What I meant is coloring the edges of a complete bipartite graph by either white or black. Say we choose two edges of $K_{(m,n)}$ and color it white and the rest black. Then the graph formed by all the vertices and all the black edges is a spanning subgraph of size $mn-2$. – chowching Feb 23 '15 at 03:14
  • I also find this statement questionable: "Every coloring of the edges will produce a spanning subgraph." – Andrew Szymczak Feb 24 '15 at 11:01
  • What I mean is if I have $K_{(m,n)}$ and I color some edges white and black with white meaning the edges colored with it are removed and black means the edges colored with it remain. So every coloring will produce a spanning subgraph. – chowching Feb 24 '15 at 14:51

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