My child graduated in mid-May, started an internship in mid-June and a job in October. We paid for most of her expenses and 2017 tuition. Can we claim her as a dependent?

[felice]

Counting days at school, travel, and at home may not be more than 6 months. Can we still claim her as a dependent because her full-time job didn't start until October?

[DoninGA]

If she is under the age of 24 and was a full time student for any 5 months in 2017 you should be able to claim her as your dependent under the Qualifying Child rules if she meets all the other requirements.

To be a Qualifying Child -

1. The child must be your son, daughter, stepchild, foster child, brother, sister, half brother, half sister, stepbrother, stepsister, or a descendant of any of them.

2. The child must be (a) under age 19 at the end of the year, (b) under age 24 at the end of the year and a full-time student or (c) any age and permanently and totally disabled.

3. The child must have lived with you for more than half of the year. Temporary absences while away at college are considered living with you.

4. The child must not have provided more than half of his or her own support for the year.

5. If the child meets the rules to be a qualifying child of more than one person, you must be the person entitled to claim the child as a qualifying child. 

6. The child must be a U.S. citizen or U.S., Canada or Mexico resident for some portion of the year.

7. The child must be younger than you unless disabled.

[Hal_Al]

Probably not. "Counting days at school, travel, and at home may not be more than 6 months" is critical. It usually hinges on  "Did she provide more than 1/2 her own support in 2017".  But in your case, the residency rule will probably govern.

Graduation year
If he/she was a student (under 24) for at least 5 months and lived with you for more than half the year, and did not provide more than 1/2 his own support for the whole year, you can still claim him. Be sure he knows you're claiming him, so he doesn't claim himself. He can only be claimed once. But, he can "file taxes" without claiming his own exemption.
The real question is who should be claiming him in this "transition" year to adulthood. You two have to agree on who is going to claim his exemption. Each should do their taxes both ways and see which way the family comes out best.  Even then, you have to meet the rules. The rule is that a child of a taxpayer can still be a “Qualifying Child” dependent, regardless of  his income, if:
1. he is a full time student under 24 for at least 5 calendar months of the year (graduating in May usually means you meet the 5 month rule)
2. he did not provide more than 1/2 his own support  (scholarships are considered 3rd party support and not support provided by the student).
3. lived with the parent (including time away at school) for more than half the year

So, it usually hinges on  "Did he provide more than 1/2 his own support in 2017".
The support value of the home you provided is the fair market rental value of the home plus utilities & other expenses divided by the number of occupants. IRS Publication 501 on page 20 has a worksheet that can be used to help with the support calculation. See: http://www.irs.gov/pub/irs-pdf/p501.pdf