Questions tagged [bipartite-graphs]

A bipartite graph is a graph whose vertices can be divided into two disjoint sets such that no two vertices in the same set are adjacent.

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1answer
134 views

At most one perfect matching of a bipartite graph

I. Given biadjacency matrix $A$ of a bipartite graph on $2n$ vertices having $n$ vertices of either color on the constraints the graph either has $0$ perfect matchings $1$ perfect matchings is it ...
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1answer
140 views

Can we construct a dessin of any genus with a cyclic automorphism group of any order?

We consider a dessin d'enfant $D$ as a bipartite graph $D$ on a complex oriented surface $S$, such that the complement $S \backslash D$ is homotopic to a collection of disks. Definition: Let an ...
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28 views

Computing bipartite matching of size $k$?

Given a bipartite graph with $n$ vertices on each side and an integer $k$, how can we compute all bipartite matchings of size $k$? The problem of computing all perfect matchings is #P-complete. But I ...
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67 views

Number of extremal $\{0,1\}$ matrices having permanent $1$ property

Is there a function which describes the number of $\{0,1\}^{n\times n}\cap\mathbb Z^{n\times n}$ matrices having permanent $1$? I think it might be $\mathsf{poly}(n!)$ bounded. Is there a function ...
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1answer
79 views

Unique bipartite perfect matchings and cycles?

Given a graph $G$ which is bipartite and balanced and has unique perfect matching let $G^{e}$ be $G$ without edge $e$. Let $G\cup G_{\pi,\pi'}$ be union of $G$ and $G_{\pi,\pi'}$ where $G_{\pi,\pi'}$ ...
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65 views

Is there a bipartite graph whose determinant corresponds to number of perfect matchings?

Let $M\in\{0,1\}^{n\times n}$ be a square integer matrix. If we consider $M$ as biadjacency of a balanced bipartite graph on $2n$ vertices having $n$ vertices of color $1$ and $n$ vertices of color $2$...
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94 views

A query on Galvin's theorem for bipartite graphs

The Galvin's theorem is the generalized version of Dinitz conjecture that states that if the maximum degree of any bipartite graph is $\Delta$, then its edges are colorable properly if each of its ...
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98 views

Two from cubic subgraph hardness

The Problem For a given graph $G$, the cubic subgraph problem asks if there is a subgraph where every vertex has degree 3. The cubic subgraph problem is NP-hard even in bipartite planar graphs with ...
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43 views

Counting the number of simple labelled bipartite graphs 𝐺𝑛,𝑚 with 𝑘 edges such that 𝑑1 vertices have degree 1

I have tried to count the number of simple labelled bipartite graphs $G_{n,m}$ with $k$ edges such that $d_1$ vertices have degree 1. Has this problem been studied? So far the only related paper I ...
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2answers
278 views

Conjecture about minimal number of edge crossings in complete bipartite graphs

I am interested in the status of the conjecture about the minimum number of edge crossings $cr(K_{m,n})$ in a drawing of the complete bipartite graph $K_{m,n}$. The Wikipedia article https://en....
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210 views

What is a bipartite hypergraph?

Bipartite graphs are very useful, and I am looking for a generalization of this concept to hypergraphs. I found two different definitions of bipartite hypergraphs: In the Wikipedia page Hypergraph, a ...
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24 views

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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216 views

Probability that a random multigraph is simple

Question. Consider a given sequence of $n$ integers $d_1$, $d_2$, $\cdots$, $d_n$ with $\sum_i d_i$ even and $d_i\le n$ for all $i$. One may sample a random multi-graph having this degree sequence ...
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2answers
90 views

One part of a bipartite graph has max degree 3. Partition the other part to 3 ~equal subsets s.t. just a fraction of first part see all 3 subsets

Let $d \gg 1$. Let $G:=(U, V, E)$ be some bipartite graph such that deg$(u) \le d$ for all $u\in U$ and deg$(v) \le 3$ for all $v \in V$. Now, is it possible to color vertices in $U$ with 3 colors ...
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56 views

If all 2-faces of a polytope are $2n$-gons, is the edge-graph bipartite?

This question on MSE has not received a satisfying answer. It can be summarized as follows: Question: Is is true that the edge-graph of a (convex) polytope is bipartite if and only if all 2-faces ...
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2answers
85 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. ...
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19 views

Hadwiger number in vertex collapse in a bipartite graph

If $G=(V,E)$ is a finite graph, let the Hadwiger number $\eta(G)$ equal the largest integer $n$ such that the complete graph $K_n$ is a minor of $G$. Is there a bipartite graph $G$ on more than $3$ ...
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47 views

How many edges can be in an unbalanced bipartite graph of girth $>6$?

Let $G = (V, E)$ be a bipartite graph with $n, m$ nodes in its bipartition and girth (shortest cycle length) $>6$. There is a simple counting argument called the Moore Bounds that gives $$|E| = O\...
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42 views

Treewidth related properties of a bipartite graph with bounded local crossing number and diameter

If a bipartite degree at most $3$ graph on $O(n^2)$ vertices with diameter at most $O(\log n)$ has property that every edge intersects at most $O(\log n)$ edges on a planar drawing then does any of ...
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1answer
79 views

Partitioning vertex set to maximize weights of inter-class edges?

An interesting problem has come up in my work, and I haven't quite been able to find references to it so I thought I'd ask here. Suppose we have some complete, weighted graph with vertex set $V$. Is ...
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1answer
73 views

Maximum number of perfect matchings in a graph of genus $g$ balanced $k$-partite graph

What is the maximum number of perfect matchings a genus $g$ balanced $k$-partite graph (number of vertices for each color in all possible $k$-colorings is within a difference of $1$) can have? I am ...
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64 views

Statistics of perfect matching and incremental perfect matchings in bipartite planar graphs?

Planar graph permanent can be reduced to determinants and so statistics should be amenable. Pick a uniformly random bipartite planar graph $G$ with $n$ vertices of each color and choose new additional ...
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1answer
289 views

Minimum planar bipartite graph to cover all perfect matching count

Given set $\mathcal T_n=\{0,1,\dots,2^n-1\}$ what is the minimum number of vertices $2m$ needed in a planar bipartite balanced graph such that at every $i\in\mathcal T_n$ there is a graph $G\in\...
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+50

Volume interpretation of number of perfect matchings in bipartite planar graphs

Permanent of biadjacency of bipartite graphs is the number of perfect matchings. In the case of planar graphs we can obtain an orientation with sign changes and get away with computing the determinant ...
5
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1answer
178 views

“König's theorem” for $T_2$-spaces?

For any topological space $(X,\tau)$ we define a matching to be a collection of non-empty and pairwise disjoint open sets. We define the matching number $\nu(X,\tau)$ to be the smallest cardinal $\...
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86 views

Have partition functions of abstract simplicial complexes been examined?

Many complicated probability distributions arising in electrical engineering and machine learning have a simple expression as a sum of products that can also be encoded in a factor graph. The ...
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1answer
113 views

An variation of an assignment problem in combinatorics: assign items to customers

Suppose we want to assign $n$ items to $m$ customers ($n \geq m$). Each assignment of an item $i$ to a customer $j$ has an associated cost $c(i,j)$. Find an assignment that maximizes the total cost. ...
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2answers
72 views

On a condition concerning the number of neighbors in bipartite graphs

For any undirected simple graph $G=(V,E)$ we define for $v\in V$ the set $N(v) = \{w\in V: \{v,w\}\in E\}$. Suppose $A, B$ are finite, disjoint sets, and $G = (A\cup B, E)$ is a bipartite graph with ...
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129 views

Is the partition of bipartite graphs NP-hard?

I wonder if the following problem is NP-hard. Is it? Given a bipartite graph $G = (U, V, E)$ with weights $w : E \to \mathbb{R}_+$, find a partition of $U$ into $U_1, U_2$ and nonempty disjoint ...
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1answer
101 views

Maximum partition of bipartite graph

Let $G = (U, V, E)$ be a bipartite graph. Let $w: E \to \mathbb{R}$ be a weight function on the edge set $E$. Given subsets $U_1,\ldots, U_k \subset U, U_i\cap U_j = \emptyset$ and a partition $V_1,\...
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1answer
440 views

Algorithm to find a $k$-partite graph

Is there any algorithm which finds any $k$-partite graph of a given graph which is known to be a $k$-partite graph? For example, you are given a graph $G$ with vertices $V$ and edges $E$, and you ...
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189 views

Dowker and neighborhood complexes: reference wanted

Let $R$ be a 0-1 matrix whose rows or columns are maximal. Q1. Is there a name for such a matrix (or, e.g., a corresponding relation)? From 0-1 matrix corresponding to an abstract simplicial ...
4
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1answer
557 views

Combinatorial optimization problem for bipartite graphs

Let $G(V_1\cup V_2, E)$ be a simple bipartite graph having $n$ vertices and $m$ edges, such that $|V_1|=|V_2|$ (which implies that $n$ is an even number). Given any node $i \in V_1\cup V_2$, we denote ...
5
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3answers
197 views

How many $40$-vertex cubic bipartite graphs have determinant $\pm 3$?

To get some feel for the size of a particular computation, I would like to know the approximate number of (pairwise-nonisomorphic) cubic bipartite graphs on $40$ vertices whose bipartite adjacency ...
2
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1answer
132 views

Existence of bipartite subgraphs satisfying degree and edge cardinality constraints

How can we prove the following conjecture? Given any simple unweighted bipartite graph $G(V_1, V_2, E)$, there always exists a subgraph $G'(V_1, V_2, E')$ of $G$ such that the two following ...
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1answer
73 views

Canonical form for a bipartite graph

I have a bipartite graph, including V1 and V2 vertices, and I would like to convert it to a canonical form. One simple method is converting this graph to a general graph by expanding its adjacency ...
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2answers
229 views

Hamiltonicity and minimal degree in bipartite graphs

Given an integer $k>1$, is there a connected bipartite graph $\Gamma = (A, B, E)$ where $A\cap B = \emptyset$ and $E \subseteq \big\{\{a, b\}:a\in A, b\in B\big\}$ such that $|A| = |B|$, $\text{...
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3answers
484 views

Hamiltonian paths in bipartite graphs with 2 sets of “almost” same cardinality

Suppose we have two finite disjoint sets $A, B \neq \emptyset$ such that $|A|$ and $|B|$ differ by at most $1$, and let $\Gamma = (A\cup B, E)$ where $E\subseteq \big\{\{a,b\}: a\in A, b\in B\big\}$ ...
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1answer
630 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 ...
2
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0answers
159 views

Minimal size of the maximum biclique

We examine a bipartite graph with two sides $R$ and $L$, and denote by $|L|$ and $|R|$ the number of nodes in each side. We know only that each vertex on side $R$ is connected to $k$ vertices on side $...
3
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1answer
145 views

Minimal size of the maximal biclique

We examine a bipartite graph with two sides $R$ and $L$, and denote by $|L|$ and $|R|$ the number of nodes in each side. We know only that each node on side $R$ is connected to $k$ nodes on side $L$, ...
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67 views

Moving subsets of bipartite graph

I am dealing with a bipartite graph $G = (L\cup R, E)$. The left and right sets are both assumed to be regular with degrees $d_L$ and $d_R$. Assume $|L| > |R|$ and hence $d_L < d_R$. Let $S \...
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1answer
134 views

Is bipartite graph genus bound by $O(\mbox{max deg})$?

We know that planar graphs have $O(1)$ degree. We know balanced (each color has same number of vertices) complete bipartite graphs have genus $O(n^2)$. If maximum and average degree are $O(n^\alpha)$ ...
6
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1answer
71 views

Looking for the name or reference regarding a bipartite graph parameter

I'm writing a paper about a math puzzle and the thing I'm studying ends up equivalent to finding the following parameter of a bipartite graph G with parts X and Y: The largest $k$ such that any $k$ ...
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2answers
762 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. ...
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59 views

Cut norm and biclique gap?

Given real $\pm1$ matrix $M\in\Bbb R^{n\times m}$ we have that cut-norm is given by $$\|M\|_C=\max_{\mathcal I\subseteq[n],\mathcal J\subseteq[m]}\Big|\sum_{(i,j)\in\mathcal I\times\mathcal J}M_{ij}\...
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141 views

Maximality with respect to having no marriage

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\...
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1answer
318 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\...
2
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1answer
89 views

Minimum number of edges to add in order to have a biclique cover

Given a bipartite graph G and a number N, what's the minimum number of edges I have to add to G in order to be able to cover the resulting graph with no more than N complete bipartite subgraphs? For ...
1
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2answers
886 views

Rank adjacency matrix bipartite graph

I am interested to know what kind of characterizations are known of the rank of bipartite graphs $G(n,m)$ ($n$ vertices on one side, $m$ on the other, $n \leq m$). When is the incidence matrix full ...