Destroying Noncomplete Regular Components in Graph Partitions

We prove that if G is a graph and r1, . . . , rk ∈ Z≥0 such that ∑k i=1 ri ≥ ∆(G) + 2 − k then V (G) can be partitioned into sets V1, . . . , Vk such that ∆(G[Vi]) ≤ ri and G[Vi] contains no non-complete ri-regular components for each 1 ≤ i ≤ k. In particular, the vertex set of any graph G can be partitioned into ⌈ ∆(G)+2 3 ⌉ sets, each of which induces a disjoint union of triangles and paths.