Can you clarify what exactly you want to compute?
Do you need a clique vertex cover or a clique edge cover? I assume you mean the former, since you said that each vertex must be present in only one clique.
This problem is equivalent to colouring the complement graph.
My package IGraph/M has minimum colouring functionality.
Now we can do this:
g = Graph[{0 <-> 1, 0 <-> 4, 0 <-> 7, 0 <-> 10, 0 <-> 14, 1 <-> 3,
1 <-> 4, 1 <-> 6, 1 <-> 7, 1 <-> 11, 1 <-> 13, 1 <-> 16, 2 <-> 5,
2 <-> 7, 2 <-> 8, 2 <-> 14, 2 <-> 15, 2 <-> 16, 3 <-> 4, 3 <-> 6,
3 <-> 12, 3 <-> 13, 3 <-> 16, 3 <-> 17, 4 <-> 5, 4 <-> 6, 4 <-> 8,
4 <-> 16, 5 <-> 7, 5 <-> 8, 5 <-> 12, 5 <-> 13, 5 <-> 16,
5 <-> 18, 5 <-> 19, 6 <-> 14, 7 <-> 14, 7 <-> 15, 7 <-> 18,
7 <-> 19, 8 <-> 9, 8 <-> 11, 8 <-> 15, 8 <-> 16, 9 <-> 12,
9 <-> 13, 9 <-> 14, 10 <-> 15, 10 <-> 19, 11 <-> 16, 11 <-> 17,
11 <-> 18, 12 <-> 19, 13 <-> 17, 13 <-> 18, 13 <-> 19, 15 <-> 16,
15 <-> 17, 15 <-> 19}, VertexLabels -> "Name"];
This is a minimum colouring of the complement graph:
In[41]:= IGMinimumVertexColoring@GraphComplement[g]
Out[41]= {5, 7, 7, 5, 1, 5, 7, 7, 2, 3, 6, 6, 6, 6, 1, 4, 3, 2, 1, 4}
IGraph/M has a function to convert a labelling of vertices with (possibly equal) consecutive integers (i.e. "membership indices") to partitions that contain vertices with the same indices.
Thus we can obtain the vertex clique cover like this:
In[42]:= cover = IGMembershipToPartitions[g]@IGMinimumVertexColoring@GraphComplement[g]
Out[42]= {{0, 7, 14}, {1, 4, 3, 6}, {10, 15, 19}, {11, 18}, {13, 17}, {16, 2, 5, 8}, {12, 9}}
This form is easy to visualize:
HighlightGraph[g,
Subgraph[g, #] & /@ cover,
GraphHighlightStyle -> "DehighlightGray"
]
Or if you like,
CommunityGraphPlot[g, cover]
This minimum cover consists of 7, not 8, cliques. If this is not what you wanted, please clarify.
