Why does 在 turn up in 全班同学在老师公布考试时间之后都紧张了起来? I have two sentences. 


总经理（在）假期结束之后立刻开始了紧张的工作.
全班同学在老师公布考试时间之后都紧张了起来.


The first sentence can eliminate 在. About the reason why 在 turns up, in my opinion, it can be the basic structure:


  Subject + When + Where + How + Verb + Object


because 老师公布考试时间 is a sentence, adding "之后" just make it indicate the time (老师公布考试时间之后). So between Subject and Time, we need a preposition. However, I cannot explain the grammar structure of the first sentence (lmao).

2 sentences may not follow the basic structure (and I misunderstood). In case I am correct:

1/ why can the first sentence be eliminated 在？ 

2/ If I am wrong, is there any other structures that match both sentences?
总经理在假期结束之后立刻开始了紧张的工作.

全班同学在老师公布考试时间之后都紧张了起来.



I don't think we should omit 在 in the first sentence either.  在... 之后 is a prepositional phrase indicating when something happens.
在 can be droppable when you put the prep phrase at the beginning of the sentence.  E. g.

(在)假期结束之后, 总经理立刻开始了紧张的工作.
(在)老师公布考试时间之后, 全班同学都紧张了起来.

There isn't any special structure involved here.  This is all about how/where you want to insert a prepositional phrase in a sentence.