Torrent Brock Biologia Dei Microrganismi Vol 1 Pdf
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E-mail Subscription to Our Newsletter Check for actual costs for usa reg, Aomori. brock biologia dei microrganismi vol 1 microbiologia generale 1 pdf.Q:
Direct proof of the connectedness of $B(n,k)$
My professor gave a direct proof to the fact that $B(n,k)$ is connected (for $n geq 2$ and $0 leq k leq n-1$) without use of prime graphs. I tried to prove it myself but I found that it is more laborious. I have already tried to prove this by contradiction.
If we assume that there exists a connected graph $G$ such that $G subseteq B(n,k)$, then, I can’t come up with a proof to the fact that $G$ is connected.
Moreover, we have the following picture to show that we can represent $B(n,k)$ by the union of two connected graphs $A(n,k)$ and $B(n,k) setminus A(n,k)$:
I have actually made my own pictures, but I don’t know how to write them down.
A:
This is based on the solution by @Or there is nothing to prove for $k=0$ and for $k=n$ because $B(n,n)cong K_n$.
Let $H$ be an induced subgraph of $B(n,k)$ with $|H|=k+1$. Since $B(n,k)$ is connected, there is a path between every pair of vertices of $H$. Choose $uin V(H)$ and $vin V(H)setminus{u}$. We have that $vin N(u)$ because $H$ is an induced subgraph of $B(n,k)$. Let $x$ be an arbitrary vertex of $H$ distinct from $u$ and $v$. We want to prove that $xin N(u)$. Choose a path from $u$ to $x$ in $H$. That path has at least $k
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