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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.
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
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
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
5
votes
What are efficient pooling designs for RT-PCR tests?
This isn't a full answer, but too long for a comment. I suppose it comes closest to trying to answer Question 3 or the general question of whether the hypercube design can be improved.
Definition Giv …
- 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
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
4
votes
1
answer
267
views
Dominating sets in subtournaments of the Paley tournament
For a tournament $T$, let $\mathrm{dom}(T)$ be the order of a smallest dominating set in $T$. Let $q$ be a prime power congruent to 3 mod 4 and let $T_q$ be the Paley tournament on $q$ vertices.
Is i …
- 1,701
4
votes
Accepted
Is König's Property for graphs inheritable from finite subgraphs?
(Just making my comment an answer as suggested.)
If every finite subgraph of $G$ satisfies Kőnig's Property, then $G$ has no odd cycles and is thus bipartite. Aharoni
(König's Duality Theorem For Infi …
- 1,701
4
votes
2
answers
252
views
Relationship between minimum vertex cover and matching width
Let $H$ be a 3-partite 3-uniform hypergraph with minimum vertex cover number $\tau(H)$ (i.e. $\tau(H)=\min\{|Q|: Q\subseteq V(H), e\cap Q\neq \emptyset \text{ for all } e\in E(H)\}$).
Question: Is $\ …
- 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
3
votes
1
answer
292
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 bipart …
- 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
3
votes
2
answers
332
views
A direct proof that every $r$-colored complete graph on $n=(r+1)m-(r-1)$ vertices has a mono...
Cockayne and Lorimer ("The Ramsey number for stripes" 1975) prove that in every $r$-colored complete graph on $n=\sum_{i=1}^rm_i+m_1-(r-1)$ vertices, where $m_1\geq \dots\geq m_r\geq 1$, has a monochr …
- 1,701
3
votes
2
answers
2k
views
Proving that every strongly connected tournament T on at least 4 vertices contains distinct ...
I have a two part question:
Is there a simple proof that every strongly connected tournament $T$ on $n\geq 4$ vertices contains distinct $u,v\in V(T)$ such that $T-u$ and $T-v$ are strongly connected …
- 1,701