A note on the not 3-choosability of some families of planar graphs

A graph G is L-list colorable if for a given list assignment L= {L(v): v ∈ V }, there exists a proper coloring c of G such that c(v) ∈ L(v) for all v ∈ V . If G is L-list colorable for any list assignment with |L(v)| k for all v ∈ V , then G is said k-choosable. In [M. Voigt, A not 3-choosable planar graph without 3-cycles, Discrete Math. 146 (1995) 325–328] and [M. Voigt, A non-3choosable planar graph without cycles of length 4 and 5, 2003, Manuscript], 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 Erdös’ relaxation of Steinberg’s conjecture to 3-choosability. © 2006 Elsevier B.V. All rights reserved.