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How many bipartite graphs can I make with m nodes on the left and k nodes on the right, given that no nodes on the left may have more than y edges, where y < k, and every node must have at least 1 edge? The graphs do not need to be connected. What is the answer in terms of m, k, and y?

Matt Chambers
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  • Can the graph be disconnected ? – Manuel Lafond Mar 02 '15 at 02:58
  • Actually yes, now that I think about it there is no requirement for the graph to be connected. Just that every m node has 1 to y edges and every k node has at least one edge. – Matt Chambers Mar 02 '15 at 03:01
  • I find this surprisingly difficult. Actually, this relates to counting the number of bipartite graphs having at least one isolated vertex (at least it's a good start to count that). But that last question has no answer as of now (see http://math.stackexchange.com/questions/35626/isolated-vertex-probabilities-for-different-random-graphs). So I'd expect this problem to be kind of hard. – Manuel Lafond Mar 02 '15 at 05:00
  • It does not depend only on $m,k,y$ unless there are additional conditions. E.g. if $m,k,y$ are all large and your graph is $K_{1,4}$ with enough isolated vertices to fill up $m,k,y$, there are 16 such subgraphs. If your graph is $K_{1,3}$ with enough isolated vertices to fill up $m,k,y$, there are 8 such subgraphs. Are you looking for upper bounds? Lower bounds? Do your subgraphs need to be spanning? – Leen Droogendijk Mar 02 '15 at 05:43
  • @Leen : to my understanding, all that's required is that the graph has no isolated nodes (OP mentions every node has at least one edge), and nodes on the left have their degree bounded by $k$. – Manuel Lafond Mar 02 '15 at 05:58
  • @Leen, no as Manuel reminded me to clarify above, the graph does not need to be connected, but all the nodes do need to be used. – Matt Chambers Mar 02 '15 at 06:15
  • The problem as stated still requires only the counted subgraphs to have no isolated vertices. As I understand now, three changes are necessary in the problem. First, the graph you start with must have no isolated vertices. Second, the subgraphs you are looking for must be spanning. Third, you need to specify what kind of boundary you are looking for, since the answer is still far from determined by $m,k$ and $y$ (e.g. take $4K_2$ and $K_{4,4}$ minus a perfect matching: both have $m=k=4$ and $y=3$, but the first has only one subgraph of the required type, the other has many). – Leen Droogendijk Mar 02 '15 at 07:51
  • Sorry for not formulating the problem in the correct formal way. Don't consider that I'm starting with any of the nodes connected (so subgraph may be the wrong term). I'm merely wanting to know the number of possible ways to connect the nodes in m and k within the constraints. – Matt Chambers Mar 02 '15 at 14:53
  • People spend time here to answer your question. You, at the very least, should spend so much time that you are able to ask the right question. You will not be able to appreciate an answer as long as you don't understand the question. – Leen Droogendijk Mar 02 '15 at 15:21
  • I didn't mean for my comment to sound snide; I really don't know how to represent this formally, so that's why I said "starting with a bipartite graph" because I don't know what to call it if you just have a set of nodes on the left and right and a set of constraints to connect them. I've rephrased the question in (I hope) a clearer way. – Matt Chambers Mar 02 '15 at 16:30
  • I think all the confusion came from the usage of the word 'subgraph'...It's true you need to watch out for the specific meaning of words - as one details may change the whole interpretation. So, if I understand, you want to know how many bipartite graphs with parts of size $m$ and $k$ there are, in which the minimum degree is $1$, and in which the left $m$ vertices have at most $y$ neighbors. – Manuel Lafond Mar 02 '15 at 16:36
  • Right. Lesson learned. :) – Matt Chambers Mar 02 '15 at 16:39

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