For questions about matchings in graph theory. A matching on a graph is a set of edges such that no two edges share a common vertex.
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Converse of Petersen's 2-Factorization Theorem
Definition: A $k$-factor of a graph is a spanning $k$-regular
subgraph.
Definition: A $k$-factorization of a graph is a partition of the edge
set into $k$-factors.
Petersen's celebrated ...
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1answer
157 views
Counting matchings in a bipartite matching-covered graph
A graph is called matching-covered if every edge is containd in a perfect matching. (Such graphs are also sometimes called "elementary", e.g. in Chapter 4 of "Matching Theory" by Lovasz & ...
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Some general theory behind lower bounds ?
Hello
Is it possible to decide whether a given or calculated lower bound is the optimal solution for a problem, like a Arc Routing Problem? If yes, when is a lower bound the optimal solution?
I just ...
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1answer
147 views
Good lower bound on matching in bipartite graph
Suppose a bipartite graph $G=(V_1 \cup V_2, E)$ is given, and one is interested in matching vertices $V_1$ to vertices $V_2$. Assume Hall's condition does not hold, so a perfect matching does not ...
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1answer
86 views
Would a graph with such maximum weighted matchings exist?
Edit Tony's answer is quite nice, but I meant something else. Sorry for editing again, I meant edges.
I am looking for a graph with 3 distinguished edges $xx'$,$yy'$,$zz'$ where ...
6
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1answer
369 views
How does this algorithmic proof of Edmonds-Gallai work?
Sorry, this is going to be technical and dirty. I am not looking for a proof of the Edmonds-Gallai structure theorem (I understand two of them, even if they are rather similar); I am trying to ...
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1answer
219 views
Degree conditions for k-factor
I am looking for a simple degree conditon that ensures the existence of a k-factor in a graph. The k is supposed to be relatively high and I don't mind the condition being a bit strict. Ideally, ...
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0answers
70 views
A non-distinct system of representative edges.
I have the following problem:
Let $ \mathcal{G} = (G_{i})\_{i} $ be a collection of graphs. I would like to find a "system of representative edges" $ f : \mathcal{G} \rightarrow \bigcup_{i} E(G_{i}) ...
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Bounds on numbers of matchings of given sizes in bipartite graphs
I am interested in the following question:
For which sets ${m_1,\ldots m_k}$ of positive integers do there exist bipartite graphs having exactly $m_i$ matchings of size $i$ for each $1\leq i \leq k$, ...
9
votes
2answers
462 views
Gale-Ryser stable marriage theorem: can we entrust matchmaking to monkeys?
Disclaimer: This is a question I have not done any real research about. I asked it myself some 5 years ago, and back then I had no idea where to start. Now I have some texts on stable matchings lying ...
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2answers
582 views
Condition on a bipartite graph to have an $m$-factor
This might be the most stupid question I am ever posting here: I am asking for a proof or a counterexample to a problem I proposed on MathLinks long ago.
Let $G$ be a bipartite graph, i. e., a graph ...
