Would a literal translation be:
"As for our school, 22nd Dec begins the release (of students) for Winter vacation"?
If so can you just say:
"...开始寒假"?
(And maybe this would move 学校 from topic to subject of the sentence?)
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Sign up to join this community放假, 放(some)假 is a fixed structure, meaning "to (canonically) begin a holiday", or "to denote the start of a holiday".
In most common examples, 我们学校十二月二十二号开始放寒假 is better translated into Our school's Winter holiday begins on December 22.
And while “开始寒假” looks like a valid word-by-word translation, it simply doesn't sound natural to a Chinese ear.
If so can you just say:
"...开始寒假"?
放 is the main verb in this sentence, 开始(begin) is the auxiliary verb, 寒假 is the noun
[我们学校][十二月二十二号][开始][寒假] would make 寒假 the main verb. Since 寒假 can only be a noun, we have to presume a verb is omitted
Example
[我们学校][十二月二十二号][开始][討論][寒假]
[我们学校][十二月二十二号][开始][取消][寒假]
[我们学校][十二月二十二号][开始][放][寒假] --> make the most sense
It is not advisable to reduce 放寒假 to 寒假
(And maybe this would move 学校 from topic to subject of the sentence?)
Change it to [我们][十二月二十二号][开始][寒假] doesn't change the fact that the main verb is omitted or you are using a noun (寒假) as a verb