Geodesic
A note on uniquely H-colourable graphs
Discussiones Mathematicae. Graph Theory, Tome 27 (2007) no. 1, pp. 39-44

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For a graph H, we compare two notions of uniquely H-colourable graphs, where one is defined via automorphisms, the second by vertex partitions. We prove that the two notions of uniquely H-colourable are not identical for all H, and we give a condition for when they are identical. The condition is related to the first homomorphism theorem from algebra.
Keywords: graph homomorphisms, core graphs, uniquely H-colourable 
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Bonato, Anthony. A note on uniquely H-colourable graphs. Discussiones Mathematicae. Graph Theory, Tome 27 (2007) no. 1, pp. 39-44. http://geodesic.mathdoc.fr/item/DMGT_2007_27_1_a3/