Past exam of the mathematics course of the University of Cambridge / 2005 / iii / Paper 10 / 1 / ii / Solution

False. The simplicial vertex boundaries need not increase below half volume. In $Q_3$,Both closed graph neighbourhoods consist of all vertices except $\{1,2,3\}$, so both have size seven. Their external vertex boundaries therefore have sizesalthough $3<4=2^{3-1}$. Adding a vertex already in the neighbourhood can reduce the external vertex boundary without enlarging the neighbourhood.