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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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Let $\mathcal{P}$ be a polyomino. Two or more rooks on $\mathcal{P}$ are called non-attacking if no path of edge-adjacent cells of $\mathcal{P}$ connects any pair of them along a row or a column. Let $...
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Thinking about this question that I report here for convenience: There are $n$ men and $n$ women. Each man has a strict preference (ordering/ranking) over all women, and each woman has a strict ...
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Definition. Let $I$ be a squarefree monomial ideal (that is, an ideal in $K[x_1,\dots,x_n]$ generated by squarefree monomials). The $k$-th squarefree power $I^{[k]}$ of $I$ is the ideal generated by ...
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Let ${M} = \{M_1, M_2, \dots, M_N\}$ be the set of all noncrossing perfect matchings on a circle with $2n$ labeled points arranged clockwise. Then $N = \frac{1}{n+1} \binom{2n}{n}$ is the $n$-th ...
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Let $G = (L, U, E)$ be a bipartite multigraph with $|L| = |U| = n$, where every vertex has degree exactly $n-1$. Since $G$ is a regular bipartite multigraph of degree $n-1$, the Birkhoff–von Neumann ...
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Suppose we have a graph, where the edges are split into blue and red edges. Is it possible to find a max weight matching such that a maximum of $n$ red edges are used?
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This is related to another question I just asked 1, after running some experiments. Let $X$ and $Y$ be two independent uniform sets of points in the unit interval, with $|X| = 2|Y|$. By running ...
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Let $X$ and $Y$ denote two sets of $m$ and $n$ points distributed uniformly at random in the unit interval. When $m$ and $n$ are both large, is there a bound for the expected cost of a minimum-weight ...
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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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The question is as stated above. I want to devise bipartite matching algorithm where it determies whether every adjacent pair of vertex on the left side of the bipartite graph has at least 1 vertex ...
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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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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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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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I am looking for an algorithm that can compute a maximum weight matching among all matchings with at least $k$ edges for some integer $k$. Note that this matching may have smaller weight than an ...
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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 ...
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I need to determine a matching with maximal weight-sum for a cycle graph with positive, negative and zero edge-weights. Question: What is the fastest way of calculating such a matching? Because of ...
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A long time ago I remember seeing a very clever construction for the following problem, but I can't find a reference for it anywhere: suppose I have a bipartite graph $G=(U\cup V, E)$, and the ...
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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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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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Question: can the solution to the dual of a Linear Program be calculated directly from the solution of the primal Linear Program? If yes, what are known algorithms and their bounds on complexity. As ...
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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'...
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I am reading Algorithms for finding K-best perfect matchings by Chegireddy and Hamacher, and I have trouble to understand their Section 2 "General algorithm for K-best perfect matchings ". ...
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(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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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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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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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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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 ...
Per Alexandersson's user avatar
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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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Suppose you have a set of objects X and a scoring function f (in which order does not matter; f(x,y) = f(y,x)) which works in the following way. Passing a viable pair of these objects to the function ...
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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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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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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, ...
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Despite some online searching I couldn't find examples of dedicated Integer Linear Programs ($\mathrm{ILP}$s) for determining smallest matchings, that are not contained in a larger one. It seems that ...
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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 ...
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Let $H=(V,E)$ be a hypergraph such that $\emptyset\notin E$. We say that $C\subseteq V$ is a (vertex) cover if for all $e \in E$ we have $C\cap e\neq \emptyset$. The minimum size that a cover can have ...
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If $\boldsymbol{A}\in\mathbb{R}^{n\times n}$ is the cost-matrix of an assignment problem, then the usual statement of the problem of finding an optimal assignment is to identify $n$ elements $a_{i,\,\...
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Let $G = (V,E)$ be a simple, undirected graph. A set $C \subseteq V$ is said to be a (vertex) cover if $C \cap e \neq \emptyset$ for all $e\in E$. A matching is a set $M\subseteq E$ of pairwise ...
Dominic van der Zypen's user avatar
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Let $H=(V,E)$ be a hypergraph. A set $M\subseteq E$ consisting of mutually disjoint members of $E$ is said to be a matching. We say $S\subseteq V$ is matchable if there is a matching $M$ such that $\...
Dominic van der Zypen's user avatar
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If $H= (V,E)$ is a hypergraph, a matching is a set $M\subseteq E$ such that $e_1\cap e_2 = \emptyset$ whenever $e_1\neq e_2 \in M$. The matching number $\mu(H)$ of a hypergraph $H=(V,E)$ with $V$ ...
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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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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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Let $G=(V_1,E_1)$ be a simple graph with vertex set $\{v_1,v_2,\ldots,v_n\}$ and let $G'=(V_2,E_2)$ be another copy of $G$ with vertex set $\{u_1,u_2,\ldots,u_n\}$. Assume $V_1\cap V_2= \emptyset$. ...
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$M$ is perfect if $M$ covers all vertices of $G$, and $M$ is extendable if $G$ has a perfect matching containing $M$. Moreover, a graph $G$ with at least $2k + 2$ vertices is said to be $k$-extendable ...
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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 $\...
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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 ...
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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 ...
Xin Zhang's user avatar
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$\textbf{Problem:}$ Find all bipartite graphs $G[X,Y]$ satisfying the following properties: $1.$ $|X|=|Y|$, where $|X|\ge 2$ and $|Y|\ge 2$. $2.$ All vertices have degree three except for two vertices ...
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given a bipartite graph $G(U,V,E\subseteq U\times V)$ with strictly positive edge-weights; is there an established name for the the task of calculating the lightest spanning subgraph and what is the ...
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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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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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