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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.
3
votes
2
answers
2k
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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 …
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
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 …
- 12.9k
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
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
1
vote
Directed version of this lemma
Regarding your Lemma 1.1, see Lemma 4.4 in Ben-Eliezer, Ido; Krivelevich, Michael; Sudakov, Benny, The size Ramsey number of a directed path, J. Comb. Theory, Ser. B 102, No. 3, 743-755 (2012). ZBL124 …
- 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 …
- 891
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 …
- 7,836
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
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
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 …
- 1,319
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
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 …
- 5,590
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