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List improper colourings of planar graphs. (English) Zbl 0940.05031
Summary: A graph $$G$$ is $$m$$-choosable with impropriety $$d$$, or simply $$(m,d)^*$$-choosable, if for every list assignment $$L$$, where $$|L(v)|\geq m$$ for every $$v\in V(G)$$, there exists an $$L$$-colouring of $$G$$ such that each vertex of $$G$$ has at most $$d$$ neighbours coloured with the same colour as itself. We show that every planar graph is $$(3,2)^*$$-choosable and every outerplanar graph is $$(2,2)^*$$-choosable. We also propose some interesting problems about this colouring.

##### MSC:
 05C15 Coloring of graphs and hypergraphs
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