All Questions
Tagged with permanent graph-theory
10 questions
2
votes
0
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60
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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 ...
- 13.9k
2
votes
0
answers
111
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Standard interpretation of permanents (of orthogonal included) over finite fields
Given a $0/1$ matrix in $\mathbb Z^{n\times n}$ the standard interpretation of permanent of the matrix is the number of perfect matchings in the underlying $2n$ vertex balanced bipartite graph with ...
- 13.9k
2
votes
0
answers
123
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Mod $2$ of $\#PM(G)$ for arbitrary G?
Permanent mod $2$ of biadjacency gives polynomial time algorithm of $\#PM(G)\mod 2$ of perfect matchings of bipartite graph. Is there a similar efficient strategy for general graphs?
- 13.9k
1
vote
0
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78
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Planar graphs with perfect matching count in linear time?
We can find Pfaffian orientation and take determinant to compute permanent in $O(n^\omega)$ time where $\omega$ is exponent of matrix multiplication.
We know that permanent of $O(n)$ vertex planar ...
- 13.9k
1
vote
1
answer
95
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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 ...
- 13.9k
28
votes
3
answers
2k
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Is every positive integer the permanent of some 0-1 matrix?
In the course of discussing another MO question we realized that we did not know the answer to a more basic question, namely:
Is it true that for every positive integer $k$ there exists a balanced ...
- 82.7k
2
votes
0
answers
64
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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: ...
- 13.9k
6
votes
0
answers
175
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Permanent bound for Laplacian matrix of signed graph
In 1986, Prof. RB Bapat shown that (see here) if $G$ is a simple connected graph on $n$ vertices, then, the permanent per$\big(L(G)\big)\ge 2(n-1)\kappa(G)$, where $L(G)$ is the Laplacian matrix of $G$...
- 640
7
votes
0
answers
349
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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 $\mathcal G_{2n}$.
...
- 13.9k
0
votes
0
answers
245
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Reduction from permanent to $(0,1)$-permanent and implication of $P \ne NP$
Valiant
shows reduction from counting the solutions of CNF formula $F$,$\#SAT(F)$
to computing permanent where $ Perm(A)= 4^{t(F)}\cdot \#SAT(F)$
for certain efficiently computable $t(F)$ and matrix $...
- 25.4k