Equitable $d$-degenerate choosability of graphs

Let ${\mathcal D}_d$ be the class of $d$-degenerate graphs and let $L$ be a list assignment for a graph $G$. A colouring of $G$ such that every vertex receives a colour from its list and the subgraph induced by vertices coloured with one color is a $d$-degenerate graph is called the $(L,{\mathcal D}_d)$-colouring of $G$. For a $k$-uniform list assignment $L$ and $d\in\mathbb{N}_0$, a graph $G$ is equitably $(L,{\mathcal D}_d)$-colorable if there is an $(L,{\mathcal D}_d)$-colouring of $G$ such that the size of any colour class does not exceed $\left\lceil|V(G)|/k\right\rceil$. An equitable $(L,{\mathcal D}_d)$-colouring is a generalization of an equitable list coloring, introduced by Kostochka at al., and an equitable list arboricity presented by Zhang. Such a model can be useful in the network decomposition where some structural properties on subnets are imposed. In this paper we give a polynomial-time algorithm that for a given $(k,d)$-partition of $G$ with a $t$-uniform list assignment $L$ and $t\geq k$, returns its equitable $(L,\mathcal{D}_{d-1})$-colouring. In addition, we show that 3-dimensional grids are equitably $(L,\mathcal{D}_1)$-colorable for any $t$-uniform list assignment $L$ where $t\geq 3$.