Improper choosability of graphs and maximum average degree

Improper choosability of planar graphs has been widely studied. In particular, Škrekovski investigated the smallest integer gk such that every planar graph of girth at least gk is k-improper 2-choosable. He proved [9] that 6 ≤ g1 ≤ 9; 5 ≤ g2 ≤ 7; 5 ≤ g3 ≤ 6 and ∀k ≥ 4, gk = 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 if l ≥ 2 then M(k, l) ≥ l + lk l+k . As a corollary, we deduce that g1 ≤ 8 and g2 ≤ 6, and we obtain new results for graphs of higher genus. We also provide an upper bound for M(k, l). This implies that for any fixed l, M(k, l) −−−→ k→∞ 2l. keywords: improper colouring, choosability, maximum average degree, planar graph, girth, genus.