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Improper choosability of graphs and maximum average degree. (English) Zbl 1104.05026
A graph $$G= (V,E)$$ is called $$k$$-improper $$l$$-choosable – or $$(l,k)^*$$-choosable – if for any list-assignment $$L$$ with $$|L(v)|\geq l$$ for each vertex $$v$$ there is a colouring of $$V$$ according to the given lists such that no vertex $$v$$ has more than $$k$$ neighbours having the same colour as $$v$$. The maximum average degree of $$G$$ is the maximum of the average degree of each of its subgraphs. This paper studies the greatest real $$M(k,l)$$ such that every graph of maximum average degree less than $$M(k,l)$$ is $$(l, k)^*$$-choosable. It is shown that $$M(k,l)\geq l+{lk\over l+k}$$, yielding as a corollary an improvement of results of R. Škrekovski [Discrete Math. 214, No. 1–3, 221–233 (2000; Zbl 0940.05027)] on the smallest integer $$g_k$$ such that every planar graph of girth at least $$k$$ is $$(2,k)^*$$-choosable: $$g_1\leq 8$$ and $$g_2\leq 6$$. Also an upper bound for $$M(k,l)$$ is given, implying that $$\lim_{k\to\infty}= 2l$$ for any fixed $$l$$. Finally, some results on improper choosability for graphs of higher genus are given.

##### MSC:
 05C15 Coloring of graphs and hypergraphs
##### Keywords:
improper coloring; planar graph; girth; genus
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