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?
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.