For any simple graph $G$, assign its nodes a weight/bit of $0$ or $1$.
Call this a *bit assignment* for $G$.
Now, generate a new bit assignment as follows:
Each node $x$'s bit is replaced by $1$ if the sum of the bits of $x$'s
neighbors is odd, and $0$ if that sum is even.
For example, the $3$-cycle $K_3$ with bits $(1,0,0)$ goes to $(0,1,1)$ and then is stable:
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&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; 
[![K3][1]][1]
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The $3$-path starting with $(1,0,0)$ goes in three steps to $(0,0,0)$ and then
is stable:
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&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; 
[![P3][2]][2]
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Say that $G$ has a *blinking assignment* if there is a bit assignment that
flips to its complement in one step, and then returns to the original assigment,
forming a cycle of length two:
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&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; 
[![Y][3]][3]
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(If animated, the graph would "blink.")

> ***Q***. Which graphs have blinking assignments?

For example, $K_3$ has no blinking assignment,
but the star $S_k$ for $k$ odd does.
Many other questions could be asked (e.g., concerning longer cycles),
but I'll focus on the above for now.

Because the update rule is to replace a node's weight with the sum
of its neighbors' weights $\bmod 2$, it seems possible this process has been
studied for some $\mathbb{Z}_n$. If so, I would appreciate a pointer.


  [1]: https://i.sstatic.net/vfIev.jpg
  [2]: https://i.sstatic.net/C51OC.jpg
  [3]: https://i.sstatic.net/Y0xMn.jpg