Improper choosability of graphs and maximum average degree

Improper choosability of planar graphs has been widely studied. In particular, Skrekovski investigated the smallest integer $g_k$ such that every planar graph of girth at least $g_k$ is $k$-improper $2$-choosable. He proved that $6\leq g_1\leq 9$; $5\leq g_2\leq 7$; $5\leq g_3\leq 6$ and $\forall k\geq 4, g_k=5$. In this paper, we study the greatest real $M(k,l)$ such that every graph of maximum average degree less than $M(k,l)$ is $k$-improper $l$-choosable. We prove that for $l\geq 2$ then $ M(k,l)\geq l+\frac{lk}{l+k}$. As a corollary, we deduce that $g_1\leq 8$ and $g_2\leq 6$. We also provide an upper bound for $M(k,l)$. This implies that for any fixed $l$, $M(k,l)\xrightarrow[k\rightarrow\infty]{}2l$.