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A note on the not 3-choosability of some families of planar graphs. (English) Zbl 1184.05048
Summary: A graph \(G\) is \(L\)-list colorable if for a given list assignment \(L = \{L(v): v\in V\}\), there exists a proper coloring \(c\) of \(G\) such that \(c(v) \in L(v)\) for all \(v\in V\). If \(G\) is \(L\)-list colorable for any list assignment with \(|L(v)| \geqslant k\) for all \(v\in V\), then \(G\) is said \(k\)-choosable. In [M. Voigt, “A not 3-choosable planar graph without 3-cycles”, Discrete Math. 146, No. 1-3, 325–328 (1995; Zbl 0843.05034)] and [M. Voigt, “A non-3-choosable planar graph without cycles of length 4 and 5”, Discrete Math. 307, No. 7-8, 1013–1015 (2007; Zbl 1112.05041)], Voigt gave a planar graph without 3-cycles and a planar graph without 4-cycles and 5-cycles which are not 3-choosable. In this note, we give smaller and easier graphs than those proposed by Voigt and suggest an extension of Erdős’ relaxation of Steinberg’s conjecture to 3-choosability.

MSC:
05C15 Coloring of graphs and hypergraphs
05C10 Planar graphs; geometric and topological aspects of graph theory
05C38 Paths and cycles
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