In a short 1946 paper "On Hamiltonian Circuits", Tutte proved the famous result that an edge in a cubic graph lies in an even number of Hamilton circuits.

He attributed the result to his friend CAB Smith, but explicitly mentioned that the proof was not Smith's proof.

What *was* Smith's proof?

mathematicalobjects and can be meaningfully non-isomorphic, a very important and modern topic. Tangentially relevant comments: (0) an article on the "CAB Smith" in the OP is this, (1) there should be people around who have known him personally, (2) the first constructive proof was given by Thomason via his 'lollipop-method', (3) the newest news on Smithian theorem is this preprint of Pitz. $\endgroup$