There were a lot more then 3 battles from 1945 to 1975. Expand on your question.
Cambodia was being invaded by both American and japenesse troops during their battles.
Vietnam
The United States did get involved in the Vietnam war because of communism in Vietnam.
Kennedy - Johnson - Nixon .
Which 3 major battles?
The UK was not involved in the Vietnam war - it was the USA!
The Gulf of Tonkin
i believe three Kennedy, Johnson, Nixon
They weren't.
Vietnam war
All three Australian armed forces and civilian formations were involved in the Vietnam war from 1962 to 1972. The related link lists them.
The answer to how many battles the United States army won in the Vietnam War is controversial. The Vietnam war was a political war that was more of an insurgency with many little ambushes and almost no actual defined battles. The most widely accepted belief about the Vietnam war is that no one won or lost.