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A directed graph is a graph with directed edges. Loops and 2-cycles are usually allowed. See also the tag *quiver*.
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 …
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
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 …
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 …
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
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 …