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Results tagged with graph-theory
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user 17798
Questions about the branch of combinatorics called graph theory (not to be used for questions concerning the graph of a function). This tag can be further specialized via using it in combination with more specialized tags such as extremal-graph-theory, spectral-graph-theory, algebraic-graph-theory, topological-graph-theory, random-graphs, graph-colorings and several others.
0
votes
Accepted
Does every primitive digraph have a directed cycle?
Yes.
For all $i,j$, $(A^k)_{ij}>0$, so there is at least one walk of length $k$ from $v_1$ to $v_2$ and there is at least one walk of length $k$ from $v_2$ to $v_1$. This closed directed walk which g …
- 1,701
4
votes
1
answer
367
views
When is it true that if $G$ is isomorphic to a spanning subgraph of $H$ and vice versa, then...
When is it true that if $G$ is isomorphic to a spanning subgraph of $H$ and $H$ is isomorphic to a spanning subgraph of $G$, then $G$ is isomorphic to $H$?
Clearly this is true if $G$ and $H$ are fini …
- 1,701
1
vote
Graph $G$ such that removing an edge leaves $G$ "unchanged"
How about a graph $G=(V,E)$ consisting of infinitely many isolated vertices and infinitely many disjoint edges. Like the random graph it has the property that for all $e\in E$, $G\simeq (V, E\setminu …
- 1,701
8
votes
1
answer
162
views
Replacing maximum degree with degeneracy in Reed's conjecture
Reed's conjecture says that $\chi(G)\leq \lceil\frac{\omega(G)+\Delta(G)+1}{2}\rceil$. One can think of $\lceil\frac{\omega(G)+\Delta(G)+1}{2}\rceil$ as the (rounded-up) average of the trivial lower …
- 1,701
2
votes
Does a strong digraph always admit a vertex that lies on some path between $\Theta(n^2)$ pai...
I don't have an answer, but I do have an idea for how a proof could go. I would make this a comment, but it is too long.
(i) Prove that every 2-strongly connected digraph (i.e. remains strongly conne …
- 1,701
6
votes
1
answer
194
views
Graphs with linear Ramsey number for two colors, but super-linear Ramsey number for three co...
Given a graph $H$, let $R_k(H)$ be the smallest integer $N$ such that in every $k$-coloring of the edges $K_N$ there is a monochromatic copy of $H$ (in other words, $R_k(H)$ is the ordinary $k$-color …
- 1,701
4
votes
1
answer
133
views
Replacing maximum degree with degeneracy in Brooks' theorem
This is related to a previous question that I asked.
The degeneracy of a graph $G$, denoted $\mathrm{degen}(G)$, is given by $\max\{\delta(H): H\subseteq G\}$. It is well known that for all graphs $G …
- 1,701
14
votes
Accepted
In an Erdős–Rényi random graph, what is the threshold for the property "every edge is contai...
I just stumbled across this question and see that it is five years old, but since I know the reference I thought I might as well share it. This threshold is determined in the paper "Local Connectivit …
- 1,701
2
votes
Perfect matchings of a regular, uniform, partite hypergraph
Here is a 3-regular 3-partite 3-uniform hypergraph with 3 vertices in each part having no perfect matching. This example generalizes in a straightforward way to give a $d$-regular $k$-partite $k$-uni …
- 1,701
0
votes
Tight bound of Turan number for K_{1,t,t}
Now to add to Jon's comment. Just take the case $t=2$ and suppose for simplicity that $n$ is divisible by 4. I'm guessing that $ex_2(n,K_{1,2,2})=\frac{n^2}{4}+\frac{n}{2}$ by taking a complete bala …
- 1,701
4
votes
A counterexample to a conjecture of Nash-Williams about hamiltonicity of digraphs?
I realize this question was asked seven years ago and hasn't had a comment in four years, but I just came across it and thought it might be worth sharing what I've learned.
As @HughThomas mentions, si …
- 1,701
4
votes
Accepted
Infinitely many counterexamples to Nash-Williams's conjecture about hamiltonicity?
These examples are symmetric digraphs, i.e. graphs. For graphs, the Nash-Williams conjecture just becomes Chvatal's theorem (If $G$ is a graph on $n\geq 3$ vertices with degree sequence $d_1\leq d_2\ …
- 1,701
1
vote
Proving that every strongly connected tournament T on at least 4 vertices contains distinct ...
After reading relep's answer I revisited the problem and came up with a different fairly simple proof for Question 1, but before I get to that, I recently found that this problem has a long history al …
- 1,701
3
votes
Induced subgraphs of the almost-disjointness graph
My first thought for the case where $|V|\leq \aleph_0$ is that surely the Rado graph can be constructed as an induced subgraph of $([\omega]^{\omega}, E)$ (since the Rado graph contains a copy of ever …
- 1,701
1
vote
Pairs of vertices with high degree difference
This is too long for a comment, but it's really just a modification of John Tuwim's answer. By using the reductions discussed in the original post, the proof becomes even simpler and it shows that $$ …
- 1,701