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For all $n\geq3$ does there always exist an $n$ vertex tournament without induced acyclic subgraphs of size $4$ or more?

Context: causal graphs in relativistic quantum information

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  • $\begingroup$ Nope. Any tournament on $2^n$ vertices contains a transitive tournament on $n+1$ vertices. (Not sharp,) Proof: easy induction on $n$. Pick any vertex in your tournament of order $2^n$; either its in-degree or its out-degree is at least $2^{n-1}$. Etc. $\endgroup$ – bof Mar 28 at 23:31
  • $\begingroup$ See OEIS A122027. $\endgroup$ – bof Mar 28 at 23:36
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Let $f(k)$ be the mimimum number of vertices for which a tournament must have an acyclic induced subgraph of size $k$.

We prove $f(k)$ exists for all $k$.

Notice $f(2)=2$.

We now prove $f(k) \leq 2f(k-1)$ for $k\geq 3$ by induction.

Suppose a graph with $2f(k-1)$ vertices exists. Take a vertex $x$ and separate the remaining vertices depending on whether the edge to $x$ goes in or out.

Notice one of these groups has size at least $f(k-1)$ and must therefore contain an acyclic induced subgraph of size $k-1$, which is still acyclic when we add $x$ to it.

It follows $f(k)\leq 2^{k-1}$.

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