Interesting pair of watches from about the same year. 1955 or thereabouts, both with the movement 453.
The one on the left looks to be a "Jubilee" model which had a reference number of 4600. If so, there would be an inscription on the outside of the back that says "Jubilee".
The one on the right has a guilloche dial pattern that is often referred to as a "tuxedo" dial, with vertical stripes. I do not know if this one had a reference number.
Not all Vacheron had reference numbers. If they did it would been on the inside of the case back. Maybe someone else can recognize the case.