Step 1: Restate the condition to check for each person.
We need persons whose festival list has neither Eid nor Onam, but does include Holi.
Step 2: Test each person one by one.
A: Holi, Dussehra, Diwali, Guru Nanak Birthday, has no Eid, no Onam, and it does have Holi, so A passes.
B: Shivratri, Christmas, Onam, Eid, has both Onam and Eid, so B fails.
C: Holi, Shivratri, Christmas, Diwali, has no Eid, no Onam, and it does have Holi, so C passes.
D: Holi, Dussehra, Guru Nanak Birthday, Eid, has Eid, so D fails.
E: Christmas, Diwali, Onam, Guru Nanak Birthday, has Onam, so E fails.
Step 3: Identify the qualifying pair.
Only A and C individually satisfy the condition of skipping Eid and Onam while joining Holi, so the correct pair is A and C.
Step 4: Rule out the other options.
A and B fails because B has Eid and Onam, C and D fails because D has Eid, and D and E fails because both D and E carry Eid or Onam.
Final Answer:
The pair is A and C. \[ \boxed{A\ and\ C} \]