Improper choosability of graphs and maximum average degree
Frédéric Havet, Jean‐Sébastien Sereni · Journal of Graph Theory · 2006
Abstract 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 article, 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 $M(k, l) \geq l + {l {\rm k} \over {l+k}}$ . 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, $M(k,l) \displaystyle\mathop{\longrightarrow}_{k \rightarrow \infty}{2l}$ . © 2006 Wiley Periodicals, Inc. J Graph Theory 52: 181–199, 2006