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.

49 questions with no upvoted or accepted answers
Filter by
Sorted by
Tagged with
8 votes
0 answers
127 views

Conceptual explanation for the gap in the spectrum of biregular trees

Ramanujan graphs are defined as $(q+1)$-regular graphs for which all nontrivial eigenvalues of the adjacency operator are contained in the interval $$[-2\sqrt{q}, 2\sqrt{q}].$$ The reason for this ...
Antoine Labelle's user avatar
8 votes
0 answers
241 views

Sum of perfect matching construction

Suppose we have two bipartite graphs $G_1$ and $G_2$ with perfect matching count $P_1$ and $P_2$ respectively then their disjoint union gives a bipartite graph with perfect matching $P_1P_2$. Is ...
Turbo's user avatar
  • 13.7k
7 votes
0 answers
346 views

Missing count in number of perfect matchings

Let $f(G)$ give number of perfect matchings of a graph $G$. Denote $\mathcal N_{2n}=\{0,1,2,\dots,n!-1,n!\}$. Denote collection of all $2n$ vertex balanced bipartite graph to be $\mathcal G_{2n}$. ...
Turbo's user avatar
  • 13.7k
6 votes
0 answers
282 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 ...
Matthieu Latapy's user avatar
6 votes
0 answers
239 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 ...
Steve Huntsman's user avatar
6 votes
0 answers
72 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 \...
WiFO215's user avatar
  • 61
4 votes
0 answers
116 views

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). ...
Dominic van der Zypen's user avatar
4 votes
0 answers
150 views

Independent sets with few neighbours

[Posted this first at math stackexchange, but it probably fits better here.] I am looking for references about the following problem. Given a (connected) bipartite graph $G$, find an independent set $...
Erik D's user avatar
  • 338
4 votes
0 answers
78 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 ...
M. Winter's user avatar
  • 12.5k
4 votes
0 answers
98 views

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 ...
Turbo's user avatar
  • 13.7k
4 votes
0 answers
150 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\...
Dominic van der Zypen's user avatar
4 votes
0 answers
207 views

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 $1\leq i \leq k$, ...
Adam 's user avatar
  • 1,327
3 votes
0 answers
88 views

Number of planar bipartite graphs

How many planar bipartite graphs are there with $m$ vertices of one color and $n$ vertices of the other color? How many non-isomorphic classes exist?
Turbo's user avatar
  • 13.7k
3 votes
0 answers
105 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 ...
Steve Huntsman's user avatar
3 votes
0 answers
100 views

Finding large bicliques in random bipartite graph

I want to find a $k$ by $r$ biclique hidden in an $M$ by $N$ random bipartite graph where edges are present with probability $p \in [0,1]$. I am specifically interested in $p \ll 1$, and large values ...
HoseinMohimani's user avatar
3 votes
0 answers
115 views

Maximum number of $4$-cycles

Suppose we have a balanced bipartite planar maximum degree $k$ graph. How many such graphs on $2n$ vertices have at most $f(n)$ maximum number of $4$ cycles for a given function $f:\Bbb R^+\...
Turbo's user avatar
  • 13.7k
3 votes
0 answers
75 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 $\...
Turbo's user avatar
  • 13.7k
2 votes
0 answers
135 views

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 $(x', y') \in M$ ...
errorist's user avatar
  • 121
2 votes
0 answers
162 views

Ramanujan graphs in Polynomial time

I am a research scholar with a computer science background, currently working on graph theory. I am working on a reduction to prove that a problem is NP-complete. I need to include the Ramanujan graph ...
Balchandar Reddy's user avatar
2 votes
0 answers
60 views

Sum of number of perfect matchings and a constant constuction

Suppose we have two bipartite graphs $G_1$ and $G_2$ with perfect matching count $P_1$ and $P_2$ respectively then their disjoint union gives a bipartite graph with perfect matching $P_1P_2$. Is ...
Turbo's user avatar
  • 13.7k
2 votes
0 answers
84 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 ...
NeoN's user avatar
  • 241
2 votes
0 answers
107 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 ...
Helene's user avatar
  • 21
2 votes
0 answers
129 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\...
GMB's user avatar
  • 1,379
2 votes
0 answers
83 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 ...
Turbo's user avatar
  • 13.7k
2 votes
0 answers
175 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 $...
Daniel Soudry's user avatar
2 votes
0 answers
122 views

Bipartite Defect in Graphs

This question is inspired by a recent related question. Given a bipartition $V=B \cup C$ of the vertices of a graph, call an edge $B$-monochromatic (or $C$-monochromatic) if both ends are in $B$ (or ...
Aaron Meyerowitz's user avatar
2 votes
0 answers
121 views

Regularity for a bipartite graph

Let $G$ be a bipartite graph with $2^n$ left vertices and $2^n$ right vertices such that: 1) degree of every vertex is not greater then $2^t$ 2) number of all edges is greater than $2^{n +t - O(\log ...
Alexey Milovanov's user avatar
2 votes
0 answers
340 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 ...
Turbo's user avatar
  • 13.7k
2 votes
0 answers
179 views

Characterize the equivalence class of bipartite graphs obtained from each other by elementary row operations on their adjacency matrices

Let $M$ be an $m\times n$ matrix real matrix. Let $G$ be a bipartite graph, with partitions $A$ and $B$, such that $|A|=m$ and $|B|=n$. A node $i\in A$ is linked to a node $j\in B$ if and only if $M_{...
valle's user avatar
  • 864
2 votes
0 answers
277 views

Bipartite independence number

Consider a balanced bipartite graph $G=(U,V,E)$, i.e., a bipartite graph with $|U|=|V|$. An independent set $I$ of $G$ is balanced if $|I \cap U| = |I \cap V|$. The bipartite independence number of $...
alezok's user avatar
  • 418
2 votes
0 answers
180 views

Details about Kelman's equivalent form of Barnette's conjecture

Barnette's conjecture states that every cubic planar bipartite 3-connected graph admits Hamiltonian cycles. Kelman claims that this conjecture is equivalent to a stronger one, which imposes some ...
Arnaud Mortier's user avatar
2 votes
0 answers
274 views

Structure of almost all bipartite graphs

I am studying some properties related to bipartite graphs and it would be useful for me to know if there is anything known about the structure of almost all bipartite graphs. For example, is it true ...
Jernej's user avatar
  • 3,433
1 vote
0 answers
77 views

Diameters of random bipartite graphs

Given two partite sets of vertices $U$ and $V$ of size $n$. Each vertex in $U$ uniformly randomly selects $K$ ($K$ is a constant and $K\ll n$) vertices in $V$ without replacement and connects a ...
Zijian Wang's user avatar
1 vote
0 answers
33 views

Tightness of the bounding the operator norm of graph by average degree from below

Let $G = (V,E)$ be a simple graph with adjacency matrix $A$. It is well known that the largest eigenvalue $\lambda_{1}$ of $A$ is contained within the interval $[d, D]$ where $d$ is the average degree ...
user135520's user avatar
1 vote
0 answers
73 views

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 ...
messi22's user avatar
  • 53
1 vote
0 answers
92 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 ...
Turbo's user avatar
  • 13.7k
1 vote
0 answers
108 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 $2$...
Turbo's user avatar
  • 13.7k
1 vote
0 answers
163 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 ...
vidyarthi's user avatar
  • 2,007
1 vote
0 answers
46 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 ...
Manfred Weis's user avatar
  • 12.6k
1 vote
0 answers
24 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 $3$ ...
Dominic van der Zypen's user avatar
1 vote
0 answers
107 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 ...
VS.'s user avatar
  • 1,816
1 vote
0 answers
138 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 ...
Thomas Edison's user avatar
1 vote
0 answers
62 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}\...
Turbo's user avatar
  • 13.7k
1 vote
0 answers
166 views

Expected number of perfect matchings in bounded degree bipartite graphs

Consider collection $\mathcal C_{n,n,\Delta}$ of every $2n$ vertex balanced bipartite graph of average degree $\Delta$. What is the expected number of perfect matching a graph in $\mathcal C_{n,n,\...
Turbo's user avatar
  • 13.7k
1 vote
0 answers
66 views

Largest number of perfect matchings in bounded genus graphs

What is the largest number of perfect matchings a genus $g$ bipartite graph on $n+m$ vertices have?
Turbo's user avatar
  • 13.7k
1 vote
0 answers
108 views

Curve associated to bipartite graph

Given real biadjacency matrix $A\in\{0,1\}^{n\times n}$ of a bipartite graph with rank $r\in[2,n-1]$, denote $A(x)$ to be matrix where $0$ is replaced by $x$ and $1$ by $1-x$. Denote $$p_1(t,x)=Det(tI-...
Turbo's user avatar
  • 13.7k
1 vote
0 answers
114 views

How many extreme maximal cliques are in an n*m 0-1 matrix?

We can use an n*m 0-1 matrix to denote a bipartite graph. Mining maximal bicliques in such matrix is an open problem. The extreme maximal clique is a special maximal clique. A clique in such matrix ...
liaomingxue's user avatar
0 votes
0 answers
65 views

Interpretation of $|V|\,|E|$ for bipartite graphs

Background: the question is motivated by a result in statistical mechanics. I am working with a generalized exclusion process with a fixed number of particles $k$ on a finite graph $G'=(V,E')$, which ...
Jens Fischer's user avatar
0 votes
0 answers
29 views

Finding a bipartite graph that contains a specific elements of perfect matchings

I am a physicist who is interested in the applications of graph theory. I've been studying the bipartite graphs and perfect matching finding problems. I see there are several research works on ...
Beom.Jean's user avatar