Combinatorica can list all $k$-colorings of the vertices of a graph $g$, which is a coloring of the vertices with no two colors adjacent, and using no more than $k$-colors:
<< Combinatorica`
<< GraphUtilities`
g = System`Graph[{1, 2, 3, 4, 5, 6}, {1 <-> 2, 3 <-> 4, 5 <-> 6}];
MinimumVertexColoring[ToCombinatoricaGraph@g, 2, All]
{{1, 2, 1, 2, 1, 2}, {1, 2, 1, 2, 2, 1}, {1, 2, 2, 1, 1, 2}, {1, 2, 2, 1, 2, 1}}
and IGraph can do the same, providing an example with minimum colors, without listing all:
IGKVertexColoring[g,2]
{{1, 2, 1, 2, 1, 2}}
Is there a way to list all the possible colourings without using Combinatorica? The IGraph package is much better and doesn't clash with system functions.