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Free choosability of outerplanar graphs. (English) Zbl 1339.05065
Summary: A graph \(G\) is free \((a,b)\)-choosable if for any vertex \(v\) with \(b\) colors assigned and for any list of colors of size \(a\) associated with each vertex \(u\neq v\), the coloring can be completed by choosing for \(u\) a subset of \(b\) colors such that adjacent vertices are colored with disjoint color sets. In this note, a necessary and sufficient condition for a cycle to be free \((a,b)\)-choosable is given. As a corollary, we obtain almost optimal results about the free \((a,b)\)-choosability of outerplanar graphs.
MSC:
05C10 Planar graphs; geometric and topological aspects of graph theory
05C15 Coloring of graphs and hypergraphs
05C38 Paths and cycles
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