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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
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