Is there a closed form formula exists to count the number of non-perfect matchings in a bipartite graph given vertices and egdes count?

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Given a bipartite graph with v vertices each side, and initially, all vertices are disconnected i.e., 0 edges were in the graph initially, how to count the number of ways of introducing v+2 or v+3 edges such that no perfect matching exists?



Any help is much appreciated. Counting each such combination seems very hard and I think that would not be the right approach as the total number of ways could be huge. Any help, topics, links, suggestions would be appreciated.




combinatorics graph-theory




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  • Two questions: your title and body seem to be asking two different questions, so which is it? Also, are these graphs labeled, for the purposes of counting?
    – Bob Krueger
    2 days ago










  • Yes, vertices are labelled, like V1, V2, v3, ...
    – user3243499
    2 days ago










  • My suggestion would be to use Konig's theorem to get a vertex cover, and then sum over edge configurations. It seems like it would have two or three nested sums though.
    – Bob Krueger
    2 days ago














up vote
1
down vote

favorite












Given a bipartite graph with v vertices each side, and initially, all vertices are disconnected i.e., 0 edges were in the graph initially, how to count the number of ways of introducing v+2 or v+3 edges such that no perfect matching exists?



Any help is much appreciated. Counting each such combination seems very hard and I think that would not be the right approach as the total number of ways could be huge. Any help, topics, links, suggestions would be appreciated.




combinatorics graph-theory




share|cite|improve this question



















  • Two questions: your title and body seem to be asking two different questions, so which is it? Also, are these graphs labeled, for the purposes of counting?
    – Bob Krueger
    2 days ago










  • Yes, vertices are labelled, like V1, V2, v3, ...
    – user3243499
    2 days ago










  • My suggestion would be to use Konig's theorem to get a vertex cover, and then sum over edge configurations. It seems like it would have two or three nested sums though.
    – Bob Krueger
    2 days ago












up vote
1
down vote

favorite









up vote
1
down vote

favorite











Given a bipartite graph with v vertices each side, and initially, all vertices are disconnected i.e., 0 edges were in the graph initially, how to count the number of ways of introducing v+2 or v+3 edges such that no perfect matching exists?



Any help is much appreciated. Counting each such combination seems very hard and I think that would not be the right approach as the total number of ways could be huge. Any help, topics, links, suggestions would be appreciated.




combinatorics graph-theory




share|cite|improve this question











Given a bipartite graph with v vertices each side, and initially, all vertices are disconnected i.e., 0 edges were in the graph initially, how to count the number of ways of introducing v+2 or v+3 edges such that no perfect matching exists?



Any help is much appreciated. Counting each such combination seems very hard and I think that would not be the right approach as the total number of ways could be huge. Any help, topics, links, suggestions would be appreciated.





combinatorics graph-theory





share|cite|improve this question










share|cite|improve this question




share|cite|improve this question









asked 2 days ago









user3243499

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  • Two questions: your title and body seem to be asking two different questions, so which is it? Also, are these graphs labeled, for the purposes of counting?
    – Bob Krueger
    2 days ago










  • Yes, vertices are labelled, like V1, V2, v3, ...
    – user3243499
    2 days ago










  • My suggestion would be to use Konig's theorem to get a vertex cover, and then sum over edge configurations. It seems like it would have two or three nested sums though.
    – Bob Krueger
    2 days ago
















  • Two questions: your title and body seem to be asking two different questions, so which is it? Also, are these graphs labeled, for the purposes of counting?
    – Bob Krueger
    2 days ago










  • Yes, vertices are labelled, like V1, V2, v3, ...
    – user3243499
    2 days ago










  • My suggestion would be to use Konig's theorem to get a vertex cover, and then sum over edge configurations. It seems like it would have two or three nested sums though.
    – Bob Krueger
    2 days ago















Two questions: your title and body seem to be asking two different questions, so which is it? Also, are these graphs labeled, for the purposes of counting?
– Bob Krueger
2 days ago




Two questions: your title and body seem to be asking two different questions, so which is it? Also, are these graphs labeled, for the purposes of counting?
– Bob Krueger
2 days ago












Yes, vertices are labelled, like V1, V2, v3, ...
– user3243499
2 days ago




Yes, vertices are labelled, like V1, V2, v3, ...
– user3243499
2 days ago












My suggestion would be to use Konig's theorem to get a vertex cover, and then sum over edge configurations. It seems like it would have two or three nested sums though.
– Bob Krueger
2 days ago




My suggestion would be to use Konig's theorem to get a vertex cover, and then sum over edge configurations. It seems like it would have two or three nested sums though.
– Bob Krueger
2 days ago















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