It's a bit hard to give a comprehensive answer without knowing exactly what sort of properties you are after, but here is a start. Such graphs are called uniquely colorable graphs (see <a href="http://mathworld.wolfram.com/Uniquelyk-ColorableGraph.html">here</a> and <a href="https://en.wikipedia.org/wiki/Uniquely_colorable_graph">here</a>). For the case of two colors they are characterized as the connected bipartite graphs, but for higher numbers of colors no such characterization is known.

In a uniquely $k$-colorable graph of $n$ vertices, the number of edges is at least $(k-1)n-\binom{k}{2}$, but otherwise they don't have to be extremely dense as your examples suggest. In fact for any $n$ there exist graphs with $n$ vertices that are uniquely 3-colorable <a href="https://www.sciencedirect.com/science/article/pii/0012365X9390220N">and have no triangles</a>.