Green’s Dictionary of Slang

(orig. Aus.) a shave.

1921		N&Q 12 Ser. IX 346: Dig-in-the-Grave. A shave.
1937		J. Curtis You’re in the Racket, Too 187: He would dive into some barber’s and have a dig in the grave.
1942		Sidney Mirror 14 Oct. in Baker (1945) 269: Struth, a bag of coke comes into the Sydney Harbour for a dig in the grave.
1953		T.A.G. Hungerford Riverslake 123: I’ve got to have a dig in the grave or the sheila’ll go crook.
1960		J. Franklyn Dict. of Rhy. Sl.
1971		J. Jones Rhy. Cockney Sl.
1980		Barltrop & Wolveridge Muvver Tongue 30: Shave: a dig in the grave.
2001		M. Coles Bible in Cockney 30: He had to have a dig-in-the-grave, wash and cut ’is Barnet.