Questions tagged [matching-theory]
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.
2
votes
1answer
43 views
Number of distinct perfect matchings/near perfect matchings in an induced subgraph
Consider a Class 1 graph with degree $\Delta\ge3$ and the induced subgraph formed by deleting a set of independent vertices of cardinality $\left\lfloor\frac{n}{\Delta}\right\rfloor$. Then, what is ...
0
votes
1answer
58 views
All even order graphs with $\Delta\ge\frac{n}{2}$ is Class 1
Are all even order graphs with maximum degree $\ge\frac{|V(G)|}{2}$ Class 1(edge-colorable(chromatic index) with $\delta(G)$ colors)? Here, $|V(G)|$ detnotes the number of vertices in the graph.
I ...
0
votes
0answers
46 views
Matching (graph theory) [closed]
There are 8 blocks, consisting of two cubes (black and white), and a platform for constructing special things. One of the blocks is glued to the platform as shown in the picture to the left. How many ...
0
votes
1answer
69 views
Hall theorem with non-saturating matching
Hall's marriage theorem states that given a bipartite graph $G=(X+Y,E)$, if there is no $X$-saturating matching, there there exists $W\subseteq X$ such that $|W|>|N_G(W)|$.
Is the following ...
6
votes
0answers
263 views
Has this notion of vertex-coloring of graphs been studied?
In a study of a quantum physics problem, I came about an apparently very natural type of vertex colorings of a graph. The colors of the vertex $v_i$ is inherited from perfect matchings $PM$ of an edge-...
2
votes
0answers
59 views
Graph pattern matching
Given a weighted, oriented, connected graph with $10^7$ vertices and $10^{10}$ edges I need to implement the algorithm for searching various patterns on this graph for less than polynomial time.
...
5
votes
0answers
152 views
Can we represent partitions by mutually parallel lines in the plane?
Lately I have become interested in the following idea: Suppose $n$ is a positive integer and $[n]=\{1,2,3,...,n\}$. Suppose we have 3 distinct partitions $b$, $g$, and $r$ of $[n]$. Assume that the ...
2
votes
1answer
138 views
Graphs with exactly $n$ perfect matchings
Is there for every $n\in\mathbb{N}$ a connected, simple, undirected graph $G_n=(V_n,E_n)$ such that $G_n$ has exactly $n$ perfect matchings?
0
votes
1answer
84 views
Connected infinite graphs in which all matchings are “small”
Is there a countable, simple, connected graph $G=(\omega, E)$ such that $\text{deg}(v)$ is infinite for all $v\in \omega$, and for all matchings $M\subseteq E$ the set $V\setminus (\bigcup M)$ is ...
3
votes
1answer
78 views
Maximal matchings in connected graphs
Let $n\in\mathbb{N}$ be a positive integer. Is there a connected graph $G$ such that $G$ cannot be coloured with less than $n$ colours, and every two maximal matchings have non-empty intersection?
(I ...
2
votes
2answers
105 views
Graphs in which all maximal matchings intersect
Let $G=(V,E)$ be a simple, undirected graph. A matching is a set $M\subseteq E$ consisting of pairwise disjoint edges. We say $M$ is maximal if it is maximal amongst all matchings in $G$ with respect ...
4
votes
2answers
73 views
Vertex cover number vs matching number
Let $G=(V,E)$ be a finite, simple, undirected graph. A matching is a set $M\subseteq E$ of pairwise disjoint edges. A vertex cover is a set $C\subseteq V$ of vertices such that $C\cap e \neq \emptyset$...
1
vote
0answers
61 views
Matchings in infinite, not necessarily bipartite, graphs
Aharoni, Nash-Williams, and Shelah have extended the famous marriage theorem for finite bipartite graphs due to Hall to arbitrary graphs.
Is there a similar generalization of Tutte's theorem on ...
1
vote
1answer
103 views
A weaker version of Dirac's theorem
This is related to Dirac's theorem.
For any finite, simple, undirected graph $G=(V,E)$ let $\delta(G)$ denote the minimal degree of all vertices.
Are there positive integers $n,c\in\mathbb{N}$ with ...
1
vote
0answers
14 views
Reducing the Dimension of Rectangular Assignment Matrices
Question:
Given a weighted complete bipartite graph $K_{m,n}(U,V,E)$, $m^2\lt n$, are there any counter examples to the assumption, that the edges of the minimum weight maximal bipartite ...
11
votes
1answer
441 views
Graphs with only disjoint perfect matchings, with coloring
The following purely graph-theoretic question is motivated by quantum mechanics.
Definitions: A bi-colored graph $G$ is an undirected graph where every edge is colored. An edge can either be ...
4
votes
1answer
214 views
Bijection between noncrossing matchings on $2b$ points and Standard Young Tableaux of size $2 \times b$
I'm currently reading a review article called Dynamical Algebraic Combinatorics: Promotion, Rowmotion, and Resonance by Jessica Striker. In this article, Striker writes that there is a ''nice'' ...
3
votes
2answers
183 views
A direct proof that every $r$-colored complete graph on $n=(r+1)m-(r-1)$ vertices has a monochromatic matching of size $m$?
Cockayne and Lorimer ("The Ramsey number for stripes" 1975) prove that in every $r$-colored complete graph on $n=\sum_{i=1}^rm_i+m_1-(r-1)$ vertices, where $m_1\geq \dots\geq m_r\geq 1$, has a ...
-2
votes
1answer
340 views
Infinite graphs with large degree but no perfect matching [duplicate]
Is there an example of an infinite connected, simple, undirected graph $G = (V,E)$ such that every vertex has $|V|$ neighbors, but $G$ does not have a perfect matching (that is, a set $M\subseteq E$ ...
3
votes
0answers
112 views
Forcing $A_1x,\dotsc,A_Kx$ to lie in a proper subspace
(This is a re-worked version of a question I asked several days ago.)
Let $J$ be the all-one matrix with $m$ rows and $n$ columns, and suppose that $J=A_1+\dotsb+A_K$ is a decomposition of $J$ into a ...
-2
votes
1answer
70 views
Matching and minimal degree
Let $n\in\mathbb{N}$ be a positive integer and let $G =(V,E)$ be a connected simple undirected graph with $|V| = 2n$. Is it true that if for the minimal degree $\delta(G)$ we have $\delta(G) \geq n$, ...
2
votes
1answer
184 views
Perfect matchings in infinite graphs
Let $G=(V,E)$ be an infinite graph such that $|V| = \kappa$ for some infinite cardinal $\kappa$, and every $v\in V$ has degree $\kappa$. Does $G$ have a perfect matching?
-4
votes
1answer
112 views
Does the Hadwiger-Nelson graph have a perfect matching?
The Hadwiger-Nelson graph on $\mathbb{R}^n$ is defined to be $(\mathbb{R}^n,E_n)$ where $$E_n = \big\{\{x,y\}: x,y\in \mathbb{R}^n \text{ and } |x-y|=1\big\},$$ where $|\cdot|$ denotes the Euclidean ...
1
vote
1answer
50 views
Hall type theorem for saturations of subsets of bipartite graphs
Let $X,Y$ be a bipartite graph and $X',Y'$ be two subsets of the vertices. Is there a Hall type theorem for the existence of matchings saturating both subsets simmultaneously?
Clearly necessary ...
3
votes
0answers
68 views
Min-Cost Max-fractional matching
I have been looking for a proof of the following statement. I think it follows from the proof of the integrality of the matching polytope but I am not so sure.
Given a bipartite graph $G=(A \cup B, ...
12
votes
2answers
295 views
Number of matchings of even cycles
By doing some calculations on the generating function of matching polynomials of cycles I made the following interesting observation:
For all positive integers $n>1$ and $k <n $, the number of ...
7
votes
1answer
337 views
Graph to Bipartite conversion preserving number of perfect matchings
Given a graph $G$ on $n$ vertices is there a technique to convert to a balanced bipartite graph $B$ with $O(n^c)$ vertices at some fixed $0<c$ in $O(n^{c'})$ time at some fixed $0<c'$ such that ...
1
vote
1answer
71 views
Tutte's Reduction of Minimum Weight d-Factors to Matching
I am currently interested in minimum weight regular d-spanners (i.e. d-factors) of complete graphs. When searching the internet for related articles, I came across this one, which is concerned with ...
5
votes
2answers
307 views
Maximum number of perfect matchings in a planar graph?
What is the maximum number of perfect matchings a planar $k$-partite $|V|$ number of vertices simple graph can have where $k=2,3,4$ ($k>4$ is impossible for a planar graph)?
Since number of ...
2
votes
0answers
55 views
Minimum size of genus $g$ bipartite graphs with $2^n$ perfect matchings
Given $n\in\Bbb Z_{\geq0}$ let $2T_{n,g}$ be size of smallest number of vertices of genus $g$ bipartite graph with $T_{n,g}$ vertices of each color such that number of perfect matchings is $2^n$.
Eg: ...
3
votes
2answers
478 views
Applications of Hafnians
I am learning about Hafnians but I am struggling to find real-world applications of them. I understand the applications of determinants, permanents, and even pfaffians but I am at a loss for Hafnians. ...
-1
votes
1answer
214 views
Marriages in infinite bipartite graphs with many neighbors
Let $A,B\neq \emptyset$ be disjoint and suppose $G = (A\cup B, E)$ is bipartite where for all $e\in E$ we have $e\cap A \neq \emptyset\neq e\cap B$. For $a\in A$ we set $N_G(a) = \{b\in B: (\exists e\...
3
votes
1answer
175 views
Induced matching number
Definition:
A $\textit{matching}$ in a graph $G$ is a subgraph consisting of pairwise disjoint edges. If the subgraph is an induced subgraph, the matching is an $\textit{induced matching}$. The ...
1
vote
1answer
159 views
Is greedy matching algorithm with normalized edge weights a 2-approximation
Given a weighted, undirected, bipartite, graph $G(V,E)$. All edge weights are assumed to be non-negative. Let $d(u)$ be the degree of vertex u. Let $c(u,v)$ be the cost of edge $(u,v)$. Goal: compute ...
8
votes
1answer
235 views
Fastest algorithm for counting perfect matchings in a general graph
Let $G(V,E)$ be a undirected graph. I am interested in the fastest known algorithm for counting the number of perfect matchings in $G(V,E)$ (which is known to be in $\#P$). In particular, what is the ...
3
votes
1answer
129 views
Unique matching completion
Assume we have bridgeless cubic graph $G(V, E)$, $n=|V|$.
By Petersen's theorem, every such graph has a perfect matching.
Moreover, given any edge in $G$ there exists a perfect matching containing ...
18
votes
2answers
1k views
Graphs with only disjoint perfect matchings
Let $G(V,E)$ be a graph. I am searching for graphs with only disjoint perfect matchings (i.e. every edge only appears in at most one of the perfect matchings).
Examples:
Cyclic graph $C_n$ with even ...
4
votes
2answers
281 views
Graphs with unique 1-Factorization
Let $G(V,E)$ be a graph with a 1-factorizations $M$ and $m=|M|$ 1-factors. I am searching for graphs with unique 1-factorizations (i.e. there is only one 1-factorization).
Examples:
Cyclic graph $...
7
votes
2answers
151 views
Do successive maximum permutations pick latin squares uniformly?
Suppose we start with a $n\times n$ matrix with entries sampled independently and uniformly at random from $[0,1]$. The weight of a set of entries will simply be the sum of those entries. A ...
8
votes
2answers
306 views
Maximum matching in a graph with no “shortcuts”
For a directed acyclic graph (DAG) $G$, denote by $G^\star$ the undirected graph obtained from $G$ by ignoring direction of its arcs. Let $e(G)=e(G^\star)$ be the number of arcs in $G$ (or edges in $G^...
3
votes
1answer
753 views
Number of perfect matchings in bipartite graph with given minimum degree
Let $G$ be a spanning subgraph of $K_{n,n}$ with minimum degree $\delta(G) \geq n/2$. It's easy to show using Hall's theorem that $G$ has a perfect matching, and the example of two disjoint copies of ...
1
vote
1answer
1k views
2-approximation algorithm for Minimum Maximal Matching (MMM) problem
I'm looking to find a 2-approximation algorithm (pseudocode) for the minimum maximal matching problem. I tried to find one but I did not manage. I want to use it to implement a program in java. Can ...
2
votes
0answers
110 views
Induced matchings in a bipartite graph with every matching having the same number of edges
Suppose $n,k$ are positive integers such that $k\mid n$.
Consider a bipartite graph $H=(A,B,E)$ such that $|A|=|B|=n$ and the edge set $E$ consists of the union of $m(H)$ induced matchings with every ...
4
votes
1answer
130 views
Are Bipartite Matching and General Matching Really Different Problems?
Questions:
Have there been attempts to either prove or disprove, that every general matching problem can be transformed into a bipartite matching problem, from whose solution the solution of ...
5
votes
2answers
280 views
Problems and Algorithms Requiring Non-Bipartite Matching
While the importance of the non-bipartite matching problem itself from an algorithmic and complexity point of view is well known, applications of non-bipartite matching are hard to find.
I did an ...
1
vote
0answers
111 views
On Schrijver's lower bound for the number of perfect matchings
Schrijver's lower bound gives the number of perfect matchings in a $k$-regular bipartite graph as $\Big(\frac{(k-1)^{k-1}}{k^{k-2}}\Big)^n$. What is the corresponding lower bound for minimum-degree $k$...
1
vote
0answers
545 views
Is there an efficient algorithm to find all the maximum matching in any tree?
A matching in a graph (G) is a set of mutually non-adjacent edges of (G). A maximum matching is a matching of maximal cardinality. A tree is an acyclic connected graph.
Is there an efficient ...
3
votes
0answers
70 views
Fraction of graphs with bound on number of perfect matchings
Asymptotically what is the fraction of balanced bipartite graph on $2n$ vertices with at most $cn^{\beta}$ edges having at most $n^\alpha$ perfect matchings for any fixed $c,\alpha>0$ and fixed $\...
1
vote
0answers
195 views
On symmetric difference of $k$-partite perfect matchings
Given a bipartite graph we know that symmetric difference of any two perfect matchings is union of even cycles.
Conversely when is it true that every union of even cycles comes from symmetric ...
4
votes
1answer
247 views
Probability bound for perfect matching
Let $p<1$ be a constant. Consider two sets $A,B$, each with $n$ vertices. For each pair $(a,b)\in A\times B$, the edge between $a$ and $b$ appears with probability $p$, independently of the ...
