If my dependent claims himself this year can I claim him next year

[KenH1]

Yes, if the 7 tests below are met.  There is no rule against "taking turns."  The only rules that matter are these ones:

1) You provided more than half of their financial support. More info

2) They made less than $4,050 in gross income during 2016 unless they are a qualifying child    (under 19 OR a full time student under 24 OR permanently disabled)

3) They live with you or they are related to you.

4) They are a U.S. citizen, resident alien, national, or a Canadian or Mexican resident.

5) They aren't (or won't be) claimed as a dependent by someone else.

6) They aren’t filing a joint return with their spouse.

7) You are not being claimed as a dependent on someone else's return.

If you do claim him as a dependent, he will still need to file his own return and he should answer Yes to "can someone else claim you as a dependent?"

[VolvoGirl]

But if he qualifies as your dependent and you do not claim him....he still can not claim himself.  He has to say he CAN be claimed on someone else's return even if you do not actually claim him.