Add smaller Hamiltonian circuits to make bigger one

Mathematics Asked by correaswebert on January 3, 2022

Is it always possible to generate a Hamiltonian circuit by adding smaller ones?

For example, a 8×8 grid can be split into four 4×4 grids, each having a Hamiltonian circuit. In this case, it is possible to combine the four circuits to make a bigger one. But is that always the case?

Also, is it necessary to have a congruent circuits to be added to make a bigger circuit? In the above example, we can find a bigger circuit by adding two 3×4 circuits, a 2×5 circuit and a 5×6 circuit also.

Note that I’m not trying to find smaller circuits in a bigger one
The above examples were randomly sketched by me, and sorry for less mathematical terms

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