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.
61 questions with no upvoted or accepted answers
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A rainbow perfect matching in an edge-colored graph with spanning color classes
This question is a sequel of my last question and is eventually motivated by recent advances in quantum physics. Given an even number $n\ge 6$ and a positive integer $k<n$, Claim from the linked ...
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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-...
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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 ...
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Hypergraphs with only disjoint perfect matchings
Let $H(n,r)$ be the set of $r$-uniform hypergraph with $n$ vertices that have only disjoint perfect matchings (i.e. every hyperedge only appears in at most one of the perfect matchings). Let $m(h(n,r))...
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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 ...
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When does a "stable" assignment of buyers into goods exist?
Consider a setting of $n$ buyers and $m$ goods.
We have a value matrix $V\in[0,1]^{n\times m}$ specifying how much each buyer values each good (everything is open information here and there is no ...
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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 ...
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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 ...
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Reorganizational matching
Motivation. My friend works in an organization that is re-organizing itself in the following somewhat laborious way: There are $n$ people currently sitting on $n$ jobs in total (everyone has one job). ...
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Max weighted matching where edge weight depends on the matching
Given a bipartite graph $G$, we seek a maximal weighted matching $E$. The particularity is below. Once an edge $e$ is chosen, the action of choosing $e$ adds a negative weight $w(e,e')$ to any other ...
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Is the size of maximum matching in vertex transitive 3-uniform hyper-graph on $n$ vertices always $\Omega(n)$?
What is the best known lower bound on the size of the maximum matching in a vertex transitive $3$-uniform hyper-graph?
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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 $1\leq i \leq k$, ...
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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 ...
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Matching of two weighted graphs allowing one-to-many mapping
I am looking for a heuristic for a graph matching problem as follows.
Given two graphs: $A$ (consisting of nodes $a_i$) and $B$ (consisting of nodes $b_i$). Typically the size of $B$ is larger than ...
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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 ...
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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, ...
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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 $\...
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A Result of Anders Bjorner: Matchings in countably infinite geometric lattices of finite height
Let $L$ be a countably infinite geometric lattice of finite height $r\ge3$. (A geometric lattice of height $r$ is an atomistic semimodular lattice such that every maximal chain has $r+1$ elements.)
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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 $(x', y') \in M$ ...
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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-...
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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 $[2k+1]$. ...
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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.
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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 ...
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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.
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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: ...
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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 ...
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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 ...
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Blossoms and Colorings
There is a striking analogy between finding maximum matchings in graphs and determining the chromatic number of graphs: both problems are fairly easy for bipartite graphs, but harder, resp. too hard ...
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Maximum cardinality general factor of a graph
Given a graph $G=(V,E)$ and a set of integers $B(v)$ associated to each vertex, a general factor of $G$ is a set of edges $F\subseteq E$ such that the degree of each vertex $v\in V$ in the graph $(V, ...
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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 ...
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How to solve a optimal matching problem where the "quality" of matching is only determined once all nodes are matched
Note: I'm not a mathematician; I'm just a biologist with basic math background trying to solve a scientific problem, so please excuse my ignorance.
The gist of the problem is as follows:
I have two ...
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Any updates on "The minimum cost perfect matching problem with conflict pair constraints"?
The subject of the paywalled article The minimum cost perfect matching problem with conflict pair constraints (MCPMPC) are perfect matchings of minimum cost that do not contain certain pairs of edges; ...
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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 $m,n,k$ ...
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Random graphs constructed by many large matchings
Let $G_{n,d}$ be $d$-regular random graph. We know that for $d \geq 3$, $G \in G_{n,d}$ a.a.s. has a $1$-factorisation when $n$ is even.
So, the resulting graph that obtained from randomly choosing $d$...
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Algorithmic complexity of calculating maximum weight $k$-regular subgraphs
Question:
what is known about the complexity of calculating the heaviest $k$-regular subgraph of a weighted symmetric graph if edge-weights can also be negative?
Please note that in contrast to $k$-...
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Path cover with sets of nodes
I am considering the following variant of the path-cover problem. I have an acyclic directed graph G=(V,E). Moreover, the set V is partitioned into $V=V_1 \cup ... \cup V_k$ (these sets are pairwise ...
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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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Concavity of expected size of a maximum matching (in a bipartite graph) w.r.t. edge probability
Given a n*n bipartite graph where each edge (between any two nodes on the opposite side) is formed i.i.d. with probability $p$, can we show a concavity result on the expected size of a maximum ...
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Topology of densest graphs whose optimal $3D$-matching can be calculated efficiently
let $A=\lbrace a_1,\,\dots,\,a_k\rbrace $ and $B=\lbrace b_1,\,\dots,\,b_{2k}\rbrace,\ A\cap B=\emptyset$ be be a partition of a graph's vertex set $V$, i.e. $V\,=\,A\cup B$.
Question:
has $G:=\...
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Do I need to find a maximum matching to get the matching number of a graph?
Let’s say we are talking about a simple undirected graph with no loops and no multiple edges. But not necessarily bipartite. And we need to find its matching number. Do we have to find a maximum ...
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Is there any theorem similar to the Tutte–Berge formula?
Tutte–Berge formula is a characterization of the size of a maximum matching in a graph.
The theorem states that the size of a maximum matching of a graph
${\displaystyle G=(V,E)}$ equals $${\...
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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 ...
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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 \...
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$1$-factorizability for linear hypergraphs with infinite edges on $\omega$
Let $H=(V,E)$ be a hypergraph. We say that $M\subseteq E$ is a matching if the members of $M$ are pairwise disjoint, and $M$ is said to be perfect if $\bigcup M = E$. Moreover, $H$ is $1$-factorizable ...
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Optimal preprocessing in the Kuhn-Munkres algorithm
The matrix formulation of the Kuhn-Munkres algorithm for solving the Linear Assignment Problem requires a preprocessing in which the minimal values of a line be subtracted from every value in that ...
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Matching generating function of 2-lifts
Let $P_G$ denote the matching generating function of a finite simple bipartite graph $G$.
Let now $H$ be a $2$-lift of $G$. We know (see for example Proposition 5.3.3 in Barvinok's book Combinatorics ...
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A simple case of a strong version of the Berge-Fulkerson conjecture
UPDATE 28 June 2019 A counterexample for Conjecture 2 has been provided. The conjecture is now demoted again to guess. The text has been updated to reflect this change, and there is now a new ...
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Perfect matchings and edge cuts in cubic graphs - part 1
Let $G$ be a bridgeless cubic (simple) graph, and let $M$ be a perfect matching in $G$. $G-M$ will necessarily be a set of circuits. For example, if we delete a perfect matching from $K_{3,3}$ we ...
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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 ...
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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 ...
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