In [**], for each non-negative integer k, we constructed a connected graph with (24)2k vertices which is uniquely 3-colorable, regular with degree k+5, and triangle-free. Here, for each positive integer n and each integer r>5, we construct a connected graph with (26)n*2(r-5) vertices which is uniquely 3-colorable, regular with degree r, and triangle-free.
References:
[*] Chong-Yun Chao, Zhibo Chen.
On Uniquely 3-colorable Graphs II,
Discrete Mathematics, July '98, 259-265.
[**] C.-Y. Chao, Z. Chen. On Uniquely 3-colorable Graphs, Discrete Math.
112 (1993) 21-27.