All Questions
Tagged with matching-theory co.combinatorics
81 questions
1
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149
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Inequalities in the classic proof of perfect matching in Erdős–Rényi graph
I am checking the classic paper by Erdős and Rényi, "On the existence of a factor of degree one of a connected random graph" with the link here. I am curious about the computation of the ...
- 97
4
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0
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113
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Find necessary & sufficient conditions for two families of sets to have $m$ pairwise disjoint common partial transversals of given sizes
Let $S$ be a finite set, $I$ a finite index set, $\mathcal A=(A_i:i\in I)$ and $\mathcal B=(B_i:i\in I)$ families of subsets of $S$.
For $J\subseteq I$, let $A(J)$ denote $\bigcup_{j\in J} A_j$.
A ...
- 1,644
3
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1
answer
140
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Generalizations of a theorem of Edmonds/Tutte on existence of a perfect matching in a graphs
It is well known that for a bipartite graph $G$ with bi-adjacency matrix $A$, then $\det A \neq 0$ (as a polynomial) iff $G$ has a perfect matching (there is a similar result for general graphs with ...
- 129
2
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0
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163
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Generalizing Hall's marriage theorem to non-perfect matchings
Let $G = (X, Y, E)$ be bipartite graph such that $|X|=|Y|=n$.
A matching $M \subseteq E$ is a subset of disjoint edges
(i.e., there does not exist a pair of edges $(x, y) \in M$ and $(x', y') \in M$ ...
- 121
0
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1
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238
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Graph alignment by considering node and edge weights
I have two complete weighted graphs, with the same number of nodes and edges. Each node has a multi-dimensional vector, which represents its features. Edge weights are float numbers between 0 to 1. I'...
- 101
1
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183
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Generalizing Hall's marriage theorem
(This question was earlier posted on stackexchange: Generalizing Hall's marriage theorem. As it received no answers there, I am reposting it here for more attention.)
Fix positive integers $m,n,k$ ...
- 410
3
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0
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231
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Counting matchings and perfect matchings
A matching in a graph is a subset of the edges such that
no two edges share a vertex. A perfect matching is a matching where every vertex is part of exactly one edge in the matching.
Counting the ...
- 15.8k
6
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296
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Catalan numbers from matchings?
There are several examples of interpreting the Catalan numbers as non-nesting or non-crossing matchings of some graph.
My question is:
Is there a family of graphs $G_1,G_2,\dotsc$ with the number of ...
- 15.8k
1
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0
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128
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Hopcroft–Karp Algorithm for a dynamic graph
As so you all know, we have Hopcroft–Karp Algorithm for maximum matching between two sides in a bipartite graph. It runs in $O(\sqrt{V} \times E)$ where $V$ is the vertex set and $E$ is the edges set.
...
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65
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Validity of an argument for an implication of NP-Completeness
Fedor Petrov has posed a notorious problem regarding the existence of a matching in this question: Resolution of multiple edges
As I see it the setting is a constrained bipartite matching and thus, ...
- 13.2k
0
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0
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110
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Bound on the number of maximum matchings in a graph
It is known that the number of perfect matchings in a graph is bounded above by the integer part of the square root of the permanent of its adjacency matrix. But, suppose I take the square root of the ...
- 2,089
7
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2
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500
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Disjoint perfect matchings in complete bipartite graph
Let $K_{n,n}$ be a complete bipartite graph with two parts $\{u_1,u_2,\ldots,u_n\}$ and $\{v_1,v_2,\ldots,v_n\}$, and let $K^-_{n,n}$ be the graph derived from $K_{n,n}$ by delete a perfect matching $\...
- 1,190
3
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1
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376
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Generalization of Marshall Hall's Theorem to non-simple bipartite graphs
Lemma 8.6.5 of the book "Matching Theory" by Lovász and Plummer states the following lemma:
Lemma: Let $G$ be a simple bipartite graph with bipartition $(A,B)$, and assume that each point ...
- 181
8
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1
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384
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Berge-Fulkerson conjecture --- the planar case
A well-known conjecture of Berge and Fulkerson says that every bridgeless cubic graph has a collection of six perfect matchings that together cover every edge exactly twice. Is this still open for ...
- 1,190
2
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0
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44
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Which edges to delete from cubic graphs to get good cycle covers?
Let $G\left(V,E\subset V\times V,\omega: V\supset \lbrace u,v\rbrace\mapsto w_{uv}\in\mathbb{R}\right);\ \left|e_{uv}\right|:=\omega_{uv}\quad$ be a cubic symmetric graph that contains a vertex-...
- 13.2k
1
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0
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61
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Algorithm for minimum weight matching with "tree topology"
Given a finite graph $G(V,E)$ with undirected and weighted edges, whose set of vertices $V$ is partitioned into a collection $\mathfrak{P}=\lbrace V_1,\,\dots,\,V_k\rbrace$ of non-empty and pairwise ...
- 13.2k
4
votes
1
answer
292
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Assignment problem with priorities and scores
I have run into a real problem that is actually a sort of assignment problem. I am describing it here because I am interested in knowing whether this problem already has a name (and whether there is ...
- 153
2
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0
answers
77
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Counting matchings in middle levels of the Boolean lattice
Let $k$ be a nonnegative integer and consider $C_k$, the set of all subsets $A$ of size $k$ in $[2k+1]=\{1,2,\ldots,2k+1\}$ as well as $C_{k+1}$, the set of all subsets $B$ of size $k+1$ in $[2k+1]$. ...
1
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0
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123
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Number of maximum matchings in bipartite graphs of positive surplus
Let $G$ be a simple bipartite graph with left part $L(G)$ and right part $R(G)$. For $S \subseteq L(G)$, denote $N(S)$ the set of neighbours of vertices of $S$. Define the surplus $s(G)$ as $\min_{S \...
- 5,590
4
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2
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257
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Relationship between minimum vertex cover and matching width
Let $H$ be a 3-partite 3-uniform hypergraph with minimum vertex cover number $\tau(H)$ (i.e. $\tau(H)=\min\{|Q|: Q\subseteq V(H), e\cap Q\neq \emptyset \text{ for all } e\in E(H)\}$).
Question: Is $\...
- 1,701
2
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0
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314
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Perfect matching in hypergraphs: tripartite, regular and unbalanced
In a balanced bipartite graph - where both sides have the same size - a sufficient condition for the existence of a perfect matching is that the graph is regular - all vertices have the same degree.
...
- 3,549
2
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0
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141
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From fractional matching to integral matching in tripartite hypergraphs
Let $G = (X\cup Y, E)$ by a bipartite graph with $n = |X|\leq |Y|$.
A fractional matching is a function assigning a non-negative weight to every edge in $E$, such that the sum of weights near each ...
- 3,549
10
votes
1
answer
2k
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What graph's minimum vertex cover equals twice the maximum matching?
Matching: https://en.wikipedia.org/wiki/Matching_(graph_theory)
Vertex Cover: https://en.wikipedia.org/wiki/Vertex_cover
It is easy to see that
$$\texttt{minimum vertex cover} \leq 2 \texttt{ ...
- 317
1
vote
1
answer
376
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Expanding Hall's theorem [closed]
I'm trying to get a "feel" about Hall's theorem and try to expand it for one to many matching.
So my question is:
Given a bipartite graph, what would be a neccessary and sufficient condition for that ...
- 19
2
votes
2
answers
223
views
Number of subgraphs with matching of size $n$ for a complete bipartite graph
Say we have a $K_{n,n}$ bipartite graph (i.e. a complete bipartite graph with $n$ nodes on each side). We induce a subgraph by deleting some subset of edges. There are $2^{n^2}$ possible subgraphs. ...
- 41
2
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1
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55
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maximum weighted matching with weights being sets
Given a set $S$ and a bipartite graph $G$, each edge $v\in E(G)$ covers a subset $S_v$ of $S$. My problem is to find a matching maximizing the number of covered elements, i.e., denote $V$ the set of ...
- 367
1
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2
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163
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Clutters with no maximum-size matchings
A clutter is a pair $C=(V,E)$ where $V\neq\emptyset$ is a set, and $E\subseteq {\cal P}(V)$ such that no member of $E$ is included in another member of $E$. A matching in $C$ is a collection of ...
- 48.9k
0
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1
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150
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Combining three matchings to form a maximal matching
Consider a regular tripartite graph $G$ with maximum degree $\Delta\ge3$ and parts $A,B,C$. Now, the induced subgraphs $A\cup B, B\cup C$ and $A\cup C$ are all bipartite.
Now, is there a way to ...
- 2,089
0
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1
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357
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A vertex transitive graph has a near perfect/ matching missing an independent set of vertices
Consider a power of cycle graph $C_n^k\,\,,\frac{n}{2}>k\ge2$, represented as a Cayley graph with generating set $\{1,2,\ldots, k,n-k,\ldots,n-1\}$ on the Group $\mathbb{Z}_n$. Supposing I remove ...
- 2,089
3
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1
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778
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Equitable edge coloring of graphs
Consider a simple regular graph $G$ with $n$ vertices and $E$ edges. Then, can we say that the edges can be colored equitably in $\Delta+1$ colors? By equitability is meant that a proper $\Delta+1$ ...
- 2,089
2
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1
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338
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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 ...
- 2,089
0
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1
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141
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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 ...
- 2,089
1
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1
answer
152
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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 ...
- 11
2
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2
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149
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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 ...
- 48.9k
5
votes
2
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910
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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 $C\cap e \neq \emptyset$...
- 48.9k
1
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0
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81
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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 ...
- 48.9k
0
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1
answer
134
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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 ...
- 48.9k
11
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1
answer
819
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 ...
- 155
4
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1
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433
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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'' ...
- 95
3
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2
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340
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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 ...
- 1,701
3
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0
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119
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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 ...
- 23k
1
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1
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79
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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 ...
- 1,835
14
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2
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481
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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 ...
- 4,484
7
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1
answer
969
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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 ...
- 13.9k
8
votes
2
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760
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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 ...
- 13.9k
2
votes
0
answers
64
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: ...
- 13.9k
6
votes
1
answer
538
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Induced matching number
Definition:
A matching in a graph $G$ is a subgraph consisting of pairwise disjoint edges. If the subgraph is an induced subgraph, the matching is an induced matching. The largest size of an induced ...
- 381
7
votes
2
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186
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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 ...
- 85.6k
15
votes
3
answers
613
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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^...
- 34.3k
3
votes
1
answer
3k
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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 ...
- 4,589