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.