Is it possible to find Hamilton path in this graph?

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I've been ask if there is any way to find an Hamilton path in this new Q4 graph

i think it's impossible because it isnt an even bipartite graph and not all the degs is even. is this true ?

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Hint:

  • This graph is bipartite.
  • If you color vertices black and white, then the difference between the counts has to be at most $1$ for the Hamilton path to exist.

I hope this helps $\ddot\smile$