There are at least 3 nonhamiltonian MO-graphs with 16 vertices.

In each case the chordless dominating cycle is 0-1-2-$\cdots$-11-0.

Then vertices 12,13,14,15 are each adjacent to 3 vertices of the cycle, thus:

(1)  2 3 4, 0 10 11, 6 7 8, 1 5 9

(2)  5 9 10, 2 3 8, 4 7 11, 0 1 6

(3)  3 4 10, 1 5 9, 2 7 8, 0 6 11

(1) means 12 is adjacent to 2, 3 and 4; 13 is adjacent to 0, 10 and 11, etc.

All of them have claws adjacent to consecutive vertices of the cycle.

I didn't check them for 3-edge colourings.

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There are at least 65 nonhamiltonian MO-graphs with 20 vertices.

All of them have claws adjacent to consecutive vertices of the cycle.

I'll search more with the extra condition tomorrow.