The study of graphs with an infinite number of vertices.
Questions tagged [infinite-graphs]
63 questions
24
votes
4 answers
Is this graph connected
Define the following graph on the vertex set ${\mathbb N}_{\geq1}\>$:
Two numbers $a$, $b\in {\mathbb N}_{\geq1}$ are connected by an edge (written $a \ \mathcal{R} \ b)$ if and only if $a+b \ | \ ab-1$.
Clearly $1$ is isolated. Can we connect all…
Free X
- 1,596
21
votes
1 answer
Cantor-Bernstein-Schröder Theorem: small proof using Graph Theory, is this correct?
The theorem:
Suppose there exist injective functions $A \to B$ and $B \to A$ between two infinite sets $A$ and $B$. Then there exists a bijection $A \to B$.
Proof:
Let $f: A \to B$ and $g: B \to A$ be injective functions.
Let $G$ be the bipartite…
WeakestTopology
- 1,197
7
votes
1 answer
Graphs with uncountably many vertices
Let $ \mathcal{H}$ be the class of all graphs with at most $ 2^{\aleph_0}$ vertices not containing a complete subgraph of size $ \aleph_1$. Show that there is no graph $ H \in \mathcal{H}$ such that every graph in $ \mathcal{H}$ is a subgraph of $…
shadow10
- 5,737
7
votes
1 answer
There exists no zero-order or first-order theory for connected graphs
Prove that no zero-order theory (i.e. propositional calculus, without quantification) or first-order theory can describe the "connected graph" (i.e. from any point one can reach each other point in finite steps).
The only weapon I know in these…
W.Rether
- 3,180
6
votes
1 answer
Killer Tree! (one of my old questions)
There is a problem that killed me! but I couldn't solve it:
We have a tree graph witch its structure is what is on image. Proof that there is no reduplicative numbers in each line.
Mahdi
- 429
6
votes
1 answer
Does there exist an infinite class two graph with no leaves?
In this question I will be talking about both finite and infinite graphs. All graphs are assumed to be simple (i.e. undirected and contain no loops or double edges) and connected. It is also assumed for the entirety of this question that every graph…
Pseudo Professor
- 573
- 3
- 13
6
votes
2 answers
If all subgraphs of two graphs are pairwise isomorphic, are the graphs themselves isomorphic?
For two graphs $G,H$ let's write $G\cong H$ if they are isomorphic.
Let's denote the set of all subgraphs of $G$ by $\mathcal S(G)$. Note that $G\in \mathcal S(G)$ and there can be elements $a,b\in \mathcal S(G)$ with $a\cong b$ and $a\neq b$ since…
SK19
- 3,266
6
votes
1 answer
Finite Unions of Dendrites
The question is a bit specific, but seems to be the most general question to ask after handling some obvious counterexamples.
Initially, I was wondering the following. Let $X$ be a one-dimensional Peano continuum. If $X$, aside from its points…
John Samples
- 481
5
votes
2 answers
Labelings of infinite directed acyclic graphs
Let $G=(V,E)$ be a countably infinite directed acyclic graph and $L$ be a finite set of vertex labels. The number $\left|V\right|$ of vertices is countable infinity and some vertices may have an infinite number of ingoing edges. I am interested in…
Jo Bain
- 73
5
votes
1 answer
Why is it so difficult to obtain the spectral properties related to infinite matrices, especially when they are not symmetric?
I believe most of the spectral theory is revolving around the bounded self-adjoint linear operators being analogous to real symmetric infinite matrices. Whereas, there are cases when the matrices are not real or symmetric. How can one find its…
Crystal
- 153
- 7
4
votes
1 answer
If every uncountable subset of an infinite graph has an infinite clique, must the graph have an uncountable clique?
Let $G$ be an uncountably infinite graph. $G$ has the property that every uncountable set of nodes includes an infinite clique. Must $G$ contain an uncountable clique?
A simple proof would be to try a Ramsey theorem-like approach to the…
Jack M
- 28,518
- 7
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- 136
4
votes
4 answers
Visual example of an infinite planar graph with degree sequence $(4^4,6^\infty)$
After reading some graph theory and talking with experts, I was intrigued. I would like to construct and visualise an infinite planar graph with degree sequence:
$$D=(4^4,6^\infty)$$
where the superscripts denote the number of vertices with that…
J. Zimmerman
- 842
4
votes
3 answers
What does $\lim\limits_{x \to \infty} f'(x)=3$ mean?
What I did was to see that the function is behaving as a straight line with gradient $3$ at infinity which implies that the function has an oblique asymptote as $x \to \infty$ but my testing portal says the answer is wrong and that instead, the…
M.Hamza Ali
- 342
3
votes
1 answer
Infinite binary tree coloring puzzle
Suppose there is an infinite single-rooted complete binary tree--i.e., every node except the root has a single parent, and every node has two children. There are also $n$ colors to choose from. The task is to paint as many full levels (sets of nodes…
alsips-cl
- 133
3
votes
1 answer
Are the theorems of P. Hall and M. Hall equivalent?
Let $A$ be a set together with and indexed collection $\{A_{i}:i\in I\}$ of (not necessarily distinct) subsets of $A$.
A system of distinct representatives of $\{A_{i}:i\in I\}$ is a collection of elements $a_{i}\in A_{i}$ for each $i\in I$ such…
John
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