All Questions
9 questions
4
votes
1
answer
193
views
Is König's Property for graphs inheritable from finite subgraphs?
Let $G = (V,E)$ be a simple, undirected graph. A set $C \subseteq V$ is said to be a (vertex) cover if $C \cap e \neq \emptyset$ for all $e\in E$. A matching is a set $M\subseteq E$ of pairwise ...
- 48.8k
3
votes
1
answer
304
views
Perfect matchings in infinite regular bipartite graphs
This question was motivated by a discussion here and is related to a previous question here.
Let $\kappa$ and $\lambda$ be cardinals such that $0<\lambda\leq \kappa$. Let $G=(A\cup B, E)$ be a ...
- 1,701
2
votes
1
answer
81
views
Are countable graphs with infinite minimal degree $1$-factorizable? [duplicate]
We say that a simple, undirected graph $G=(V,E)$ is $1$-factorizable if there is a partition of $E$ such that every member of the partition is a perfect matching of $G$. It is easy to see that any $1$-...
- 48.8k
5
votes
1
answer
125
views
Generalization of Menger's theorem to infinite graphs
Aharoni and Berger generalized Menger's Theorem to infinite graphs: For any digraph, and any subsets $A$ and $B$, there is a family $F$ of disjoint paths from $A$ to $B$ and a set separating $B$ from $...
- 1,644
0
votes
1
answer
92
views
Connected infinite graphs in which all matchings are "small"
Is there a countable, simple, connected graph $G=(\omega, E)$ such that $\text{deg}(v)$ is infinite for all $v\in \omega$, and for all matchings $M\subseteq E$ the set $V\setminus (\bigcup M)$ is ...
- 48.8k
-2
votes
1
answer
362
views
Infinite graphs with large degree but no perfect matching [duplicate]
Is there an example of an infinite connected, simple, undirected graph $G = (V,E)$ such that every vertex has $|V|$ neighbors, but $G$ does not have a perfect matching (that is, a set $M\subseteq E$ ...
- 48.8k
-1
votes
1
answer
378
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\...
- 48.8k
0
votes
1
answer
338
views
Stable marriages for infinite bipartite graphs
Short and informal version: Does the stable marriage problem have a solution if there are $\kappa$ men and $\kappa$ women for any cardinal $\kappa \geq \aleph_0$?
Long and formal version: Let $\kappa$...
- 48.8k
3
votes
2
answers
388
views
Maximum matchings in infinite graphs
For any graph $G=(V,E)$ we define $\mu(G) = \sup\{|M|: M\subseteq E(G) \text{ is a matching}\}$.
Is there a graph $G=(V,E)$ such that for every matching $M\subseteq E$ we have $|M|<\mu(G)$?
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