All Questions
16 questions with no upvoted or accepted answers
32
votes
0
answers
3k
views
Vertex coloring inherited from perfect matchings (motivated by quantum physics)
Added (19.01.2021): Dustin Mixon wrote a blog post about the question where he reformulated and generalized the question.
Added (25.12.2020): I made a youtube video to explain the question in detail.
...
- 155
8
votes
0
answers
245
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 ...
- 13.9k
7
votes
0
answers
203
views
Upper bound on the number of perfect matchings in $K_{3,3}$-free graphs
Let $G=(V,E)$ be a graph with an even number of vertices $|V|=2n$. Assume that $G$ is $K_{3,3}$-free i.e. it does not contain a graph isomorphic to biclique $K_{3,3}$. A perfect matching of $G$ is a ...
- 965
7
votes
0
answers
349
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 $\mathcal G_{2n}$.
...
- 13.9k
6
votes
0
answers
154
views
Complexity of finding three perfect matchings with no edge in common in a bridgeless cubic graph
According to a conjecture:
Conjecture (Fan & Raspaud, 1994) Every bridgeless cubic graph contains three perfect matchings with no edge in common.
Equivalent statement here
Main question:
...
- 25.4k
3
votes
0
answers
232
views
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 ...
- 15.8k
2
votes
0
answers
124
views
Symmetric matching in special graphs
Let $G$ be a bipartite graph, $L$ ($R$) be the set of vertices in the left (right) part.
Consider a graph $T$ with the set of vertices $R \times L$ ( $L \times R$ ) in the left (right) part. For any $...
2
votes
0
answers
163
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 $(x', y') \in M$ ...
- 121
2
votes
0
answers
108
views
Counting number of perfect matchings
Counting perfect matchings in bipartite graphs is $\# P$ complete. Let $G(V,E)$ be a graph known to have $d$ number of perfect matchings. Bipartite it the obvious way by adding $E$ vertices with one ...
- 13.9k
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 ...
- 13.9k
2
votes
0
answers
64
views
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
1
vote
0
answers
129
views
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.
...
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?
- 13.9k
0
votes
0
answers
84
views
Bounds for smallest non-trivial designs
Given $s>t\ge 2$, let $N(s,t)$ be the smallest integer $n>s$ such that there exists an “$(n;s;t;1)$-design” (i.e., a collection of $s$-subsets $e_1,\dots,e_m$ of $[n]:=\{1,\dots,n\}$, such that ...
- 3,499
0
votes
0
answers
110
views
Bound on the number of maximum matchings in a graph
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 ...
- 2,089
0
votes
0
answers
54
views
Do all induced subgraphs of powers of cycles have a perfect matching
Do all independence induced subgraphs of powers of cycles have a distinct 1-factor? By independence induced, I mean those induced subgraphs which are formed by removing a maximal independent set of ...
- 2,089