On Vertex Partitions and the Colin de Verdière Parameter

We study vertex partitions of graphs according to their Colin de Verdiere parameter μ. By a result of Ding et al. [DOSOO] we know that any graph G with μ ( G ) ⩾ 2 admits a vertex partition into two graphs with μ at most μ ( G ) − 1 . Here we prove that any graph G with μ ( G ) ⩾ 3 admits a vertex partition into three graphs with μ at most μ ( G ) − 2 . This study is extended to other minor-monotone graph parameters like the Hadwiger number.