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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5
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2answers
243 views

Matching number and chromatic number

If $G$ is a (finite) graph, denote with $\mu(G)$ the size of any maximum matching in $G$ (this number is also called the "matching number" of $G$). For odd integers $n$ we have $n=\chi(K_n) = ...
5
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2answers
202 views

Roots of matching polynomial of graph

At the end of this preprint, I make the following conjecture concerning the roots of the matching polynomial: If a graph $G$ is connected and contains a cycle, then the spectral radius of $G$ ...
3
votes
0answers
74 views

Counting perfect matchings with integrals

Has anyone used the Joni-Rota-Godsil integral formula (see details below) to count perfect matchings of square-grid graphs, Aztec diamond graphs, hexagon-honeycomb graphs, etc.? (Or even just to ...
0
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1answer
100 views

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 ...
0
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1answer
212 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 & ...
0
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1answer
166 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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vote
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
votes
1answer
378 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 ...
3
votes
1answer
235 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, ...
2
votes
0answers
73 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}) ...
4
votes
0answers
146 views

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
470 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 ...
6
votes
2answers
636 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 ...