Question

In a simple graph with $2m$ vertices and a unique perfect matching, prove that $|E(G)|$ is bounded by $m^2$.

score 80 · Answer 1 · 2023-11-30 20:23:12

80

Views

In a simple graph with $2m$ vertices and a unique perfect matching, prove that $|E(G)|$ is bounded by $m^2$.

Published on 30 Nov 2023 - 20:23

score 55 · Answer 2 · 2023-12-07 15:42:40

55

Views

Notion of neighborhood in Hall's Theorem

Published on 07 Dec 2023 - 15:42

score 76 · Answer 3 · 2026-03-26 04:28:31

76

Views

Multiset Matching

Published on 26 Mar 2026 - 4:28

score 73 · Answer 4 · 2026-03-25 23:42:34

73

Views

G a k-connected bipartite graph with bipartition $(A, B)$ such that $|A|$, $|B| \geq 2k$. Show that $G$ contains a matching of size $2k$.

Published on 25 Mar 2026 - 23:42

score 92 · Answer 5 · 2023-12-20 05:42:10

92

Views

Maximise $\sum_{x,y\in S} GCD(x,y)$ where $S = \{1,2,\ldots, 100\}$ and each number appears once in the sum

Published on 20 Dec 2023 - 5:42

score 42 · Answer 6 · 2024-01-02 13:32:23

42

Views

Stable matching theorem for non-equal sides

Published on 02 Jan 2024 - 13:32

score 86 · Answer 7 · 2026-03-25 19:05:46

86

Views

Vertices of Matchings in bipartite Graph are Matroid

Published on 25 Mar 2026 - 19:05

score 49 · Answer 8 · 2024-01-03 02:19:26

49

Views

Prove that $P_{\text {match }}(G) \cap\left\{x: 1^T x=k\right\}$ is the convex hull of all matchings in $G$ of size exactly $k$.

Published on 03 Jan 2024 - 2:19

score 91 · Answer 9 · 2024-01-03 17:54:45

91

Views

Prove that G-x-y has a perfect matching iff x and y are on opposite sides where G is Hamiltonian

Published on 03 Jan 2024 - 17:54

score 27 · Answer 10 · 2024-01-05 13:53:06

27

Views

Find correspondences between a set of ellipsoids and a set of ellipses

Published on 05 Jan 2024 - 13:53

score 88 · Answer 11 · 2024-01-13 19:20:02

88

Views

Graph theory (Perfect matchings)

Published on 13 Jan 2024 - 19:20

score 64 · Answer 12 · 2024-01-17 22:57:33

64

Views

Counting the number of "permitted" pairings between two sets

Published on 17 Jan 2024 - 22:57

score 43 · Answer 13 · 2024-01-20 10:04:24

43

Views

Find bipartite graph with properties

Published on 20 Jan 2024 - 10:04

score 46 · Answer 14 · 2026-03-26 04:53:48

46

Views

If the bridges of a $3$-regular graph $G$ lie on a single path then $G$ has a $1$-Factor (perfect matching)

Published on 26 Mar 2026 - 4:53

score 84 · Answer 15 · 2024-01-22 18:31:21

84

Views

Finding a perfect matching that include $e_{1}$ and exclude $e_{2}$ in a connected bipartite graph

Published on 22 Jan 2024 - 18:31

List Question

In a simple graph with $2m$ vertices and a unique perfect matching, prove that $|E(G)|$ is bounded by $m^2$.

Notion of neighborhood in Hall's Theorem

Multiset Matching

G a k-connected bipartite graph with bipartition $(A, B)$ such that $|A|$, $|B| \geq 2k$. Show that $G$ contains a matching of size $2k$.

Maximise $\sum_{x,y\in S} GCD(x,y)$ where $S = \{1,2,\ldots, 100\}$ and each number appears once in the sum

Stable matching theorem for non-equal sides

Vertices of Matchings in bipartite Graph are Matroid

Prove that $P_{\text {match }}(G) \cap\left\{x: 1^T x=k\right\}$ is the convex hull of all matchings in $G$ of size exactly $k$.

Prove that G-x-y has a perfect matching iff x and y are on opposite sides where G is Hamiltonian

Find correspondences between a set of ellipsoids and a set of ellipses

Graph theory (Perfect matchings)

Counting the number of "permitted" pairings between two sets

Find bipartite graph with properties

If the bridges of a $3$-regular graph $G$ lie on a single path then $G$ has a $1$-Factor (perfect matching)

Finding a perfect matching that include $e_{1}$ and exclude $e_{2}$ in a connected bipartite graph

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