Take $\mathbb{Z}[i]\pmod{4-i,1+4i}$. It has seventeen elements. Each element $m+ni$ has eight neighbours, that differ from it by $1,-1,i,-i,1+i,1-i,-1+i,-1-i$.
Two copies of this torus can be coloured with the same seventeen colours so that any two colours are neighbours in one of the tori. The colour at $m+ni$ in the first torus goes to $2m+2ni$ in the second torus.
Three torii of area 25, though different complex periods in this example, can each be coloured in 25 colours so that any two colours are neighbours in one of the torii. I can't find four torii of area 33?
4 7 10 13 16 19 22 8 11 14 17 20 23 1 12 15 18 21 24 2 5 16 19 22 0 3 6 9 20 23 1 4 7 10 13 24 2 5 8 11 14 17 3 6 9 12 15 18 21
11 1 16 6 21 11 1 9 24 14 4 19 9 24 7 22 12 2 17 7 22 5 20 10 0 15 5 20 3 18 8 23 13 3 18 1 16 6 21 11 1 16 24 14 4 19 9 24 14
18 7 21 10 24 13 2 23 12 1 15 4 18 7 3 17 6 20 9 23 12 8 22 11 0 14 3 17 13 2 16 5 19 8 22 18 7 21 10 24 13 2 23 12 1 15 4 18 7