Using the double question mark to see the source code (`vertex_coloring??`

) I see that it loops over increasingly large values of `k`

, testing whether the graph is `k`

-colorable:

```
while True:
# tries to color the graph, increasing k each time it fails.
tmp = vertex_coloring(g, k=k, value_only=value_only,
hex_colors=hex_colors, verbose=verbose)
if tmp is not False:
if value_only:
return k
else:
return tmp
k += 1
```

So one simple idea is to do this loop manually. If the process quits, at least you'll know what values of `k`

have already been checked:

```
sage: from sage.graphs.graph_coloring import vertex_coloring
sage: from sys import stdout
sage: g = graphs.CompleteGraph(9)
sage: for k in range(15):
print '%s-colorable: %s'%(k,vertex_coloring(g, k=k, value_only=True))
stdout.flush()
....:
0-colorable: False
1-colorable: False
2-colorable: False
3-colorable: False
4-colorable: False
5-colorable: False
6-colorable: False
7-colorable: False
8-colorable: False
9-colorable: True
10-colorable: True
11-colorable: True
12-colorable: True
13-colorable: True
14-colorable: True
```

I suspect (but don't know) that you will need to write your own vertex_coloring function. Copy the source of the built-in function and modify it so that it saves whatever it has calculated every now and then.

Thanks. It seems the only possible way.