Equitable tree colouring of graphs

Let $k \in \mathbb{N}$ and let $G$ be a simple graph with maximum degree $\Delta$. A $k$-colouring $\varphi$ of $G$ is an assignment of colours from $\{1,2,\ldots,k\}$ to the vertices of $G$. We call $\varphi$ proper if adjacent vertices receive distinct colours, and equitable if the sizes of any two colour classes differ by at most one. The celebrated Hajnal--Szemer\'{e}di theorem states that a proper equitable $k$-colouring exists whenever $k \ge \Delta + 1$. In this paper, we study its tree colouring variant in which each colour class induces a forest. This is closely related to the vertex arboricity which was introduced by Chartrand, Kronk, and Wall. More precisely, we prove that if $n \ge 3\Delta^4$ and $k \ge (\Delta+2)/2$, then every $n$-vertex graph with maximum degree at most $\Delta$ contains an equitable tree $k$-colouring. This confirms a conjecture of Wu, Zhang, and Li when $\Delta$ is even and up to an additive constant of $1$ otherwise for large $n$. We also consider $d$-degenerate colouring in which each colour class induces a $d$-degenerate graph.