 Solution

False. In the grid graph $[9{]}^2$, take the four-row stripIt has $36=6^2$ vertices, and only the nine edges between rows four and five lie in its edge boundary. The corner square $[6{]}^2$ has six boundary edges along each of its two exposed sides, for a total of twelve. Thus a set of size $t^2$ can have smaller edge boundary than $[t{]}^2$.