Here's a family of counterexamples. Start with a k-clique. For each node in this clique, connect it to each node of a fresh copy of K_{k-1,k-1}, i.e., the complete bipartite graph on k-1 plus k-1 nodes. For every permanent node in the clique, the residual graph is its copy of K_{k-1,k-1} and the clique. The nodes in K_{k-1,k-1} have degree k and the other clique nodes have degree k - 1, so the latter get deleted.
Here's a 16-node counterexample, obtained by setting k = 4 and identifying parts of the K_{3,3}s in a ring:
{0: {1, 2, 3, 4, 5, 6, 7, 8, 9},
1: {0, 2, 3, 7, 8, 9, 10, 11, 12},
2: {0, 1, 3, 10, 11, 12, 13, 14, 15},
3: {0, 1, 2, 4, 5, 6, 13, 14, 15},
4: {0, 3, 7, 8, 9, 13, 14, 15},
5: {0, 3, 7, 8, 9, 13, 14, 15},
6: {0, 3, 7, 8, 9, 13, 14, 15},
7: {0, 1, 4, 5, 6, 10, 11, 12},
8: {0, 1, 4, 5, 6, 10, 11, 12},
9: {0, 1, 4, 5, 6, 10, 11, 12},
10: {1, 2, 7, 8, 9, 13, 14, 15},
11: {1, 2, 7, 8, 9, 13, 14, 15},
12: {1, 2, 7, 8, 9, 13, 14, 15},
13: {2, 3, 4, 5, 6, 10, 11, 12},
14: {2, 3, 4, 5, 6, 10, 11, 12},
15: {2, 3, 4, 5, 6, 10, 11, 12}}