$(1, k)$-coloring of graphs with girth at least $5$ on a surface

A graph is $(d_1, ..., d_r)$-colorable if its vertex set can be partitioned into $r$ sets $V_1, ..., V_r$ so that the maximum degree of the graph induced by $V_i$ is at most $d_i$ for each $i\in \{1, ..., r\}$. For a given pair $(g, d_1)$, the question of determining the minimum $d_2=d_2(g; d_1)$ such that planar graphs with girth at least $g$ are $(d_1, d_2)$-colorable has attracted much interest. The finiteness of $d_2(g; d_1)$ was known for all cases except when $(g, d_1)=(5, 1)$. Montassier and Ochem explicitly asked if $d_2(5; 1)$ is finite. We answer this question in the affirmative with $d_2(5; 1)\leq 10$; namely, we prove that all planar graphs with girth at least $5$ are $(1, 10)$-colorable. Moreover, our proof extends to the statement that for any surface $S$ of Euler genus $\gamma$, there exists a $K=K(\gamma)$ where graphs with girth at least $5$ that are embeddable on $S$ are $(1, K)$-colorable. On the other hand, there is no finite $k$ where planar graphs (and thus embeddable on any surface) with girth at least $5$ are $(0, k)$-colorable.