3-Choosability of Triangle-Free Planar Graphs with Constraints on 4-Cycles

Authors:
Zdeněk Dvořák;Bernard Lidický;Riste Škrekovski
Affiliations:
rakdver@kam.mff.cuni.cz and bernard@kam.mff.cuni.cz;-;riste.skrekovski@FmF.uni-lj.si
Venue:
SIAM Journal on Discrete Mathematics
Year:
2010

Citing 5
Cited 1

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A not 3-choosable planar graph without 3-cycles

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RETRACTED ARTICLE: On 3-choosability of planar graphs with neither adjacent triangles nor 5-, 6- and 9-cycles

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Abstract

A graph is $k$-choosable if it can be colored whenever every vertex has a list of at least $k$ available colors. We prove that if a triangle-free planar graph is not 3-choosable, then it contains a 4-cycle that intersects another 4- or 5-cycle in exactly one edge. This strengthens Thomassen's result [C. Thomassen, J. Combin. Theory Ser. B, 64 (1995), pp. 101-107] that every planar graph of girth at least 5 is 3-choosable. In addition, this implies that every triangle-free planar graph without 6- and 7-cycles is 3-choosable.