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Can we draw a graph with only 9 vertices with only cycles of length 5 through 9 ?

Draw a connected graph having exactly 9 vertices that has at least one cycle of each length from 5 through 9, but has no cycles of any other length.

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  • $\begingroup$ Try it yourself first. Draw a $9$-vertex graph with a $9$-cycle, try adding some more edges to it to get the other cycles, and if you get stuck, then it will do you any good to ask for help. $\endgroup$ – Misha Lavrov Oct 25 '18 at 20:13
  • $\begingroup$ It is easy for me to draw a 9-vertex graph with cycles of length 5,6,7,9. But I can not find a cycle of length 8. $\endgroup$ – Reza Fallah Moghaddam Oct 25 '18 at 20:17
  • $\begingroup$ Hm. Maybe it is not possible. $\endgroup$ – Misha Lavrov Oct 25 '18 at 20:35
  • $\begingroup$ can we prove it? It seems that it is impossible. $\endgroup$ – Reza Fallah Moghaddam Oct 25 '18 at 20:38
  • $\begingroup$ There's the casework approach I've outlined below. Maybe there's another argument possible, too. $\endgroup$ – Misha Lavrov Oct 25 '18 at 20:44
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Both trying to find such a graph and proving that it does not exist can be done by casework.

Start with the $9$-cycle, since there is only one way it can look. We need to add more edges to get more cycle lengths, and all options so far are equivalent: we connect two vertices at distance $4$ around the cycle.

If we add an arbitrary one of these edges, then several other options can be eliminated since they'd create a short cycle. All our remaining choices are drawn in red below:

edge options cycles

Now, it is easy to check all the remaining cases, and none of them produce all five cycle lengths we want.

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