3-choosability of Triangle-free Planar Graphs with Constraint on 4-cycles
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چکیده
A graph is k-choosable if it can be colored whenever every vertex has a list of at least k available colors. A theorem by Grötzsch [2] asserts that every triangle-free planar graph is 3-colorable. On the other hand Voigt [10] gave such a graph which is not 3-choosable. We prove that every triangle-free planar graph such that 4-cycles do not share edges with other 4and 5-cycles is 3-choosable. This strengthens the Thomassen’s result [8] that every planar graph of girth at least 5 is 3-choosable. In addition, this implies that every triangle-free planar graph without 6and 7-cycles is 3-choosable.
منابع مشابه
3-Choosability of Triangle-Free Planar Graphs with Constraints on 4-Cycles
A graph is k-choosable if it can be colored whenever every vertex has a list of at least k available colors. A theorem by Grötzsch [2] asserts that every triangle-free planar graph is 3-colorable. On the other hand Voigt [10] found such a graph which is not 3-choosable. We prove that if a triangle-free planar graph is not 3-choosable, then it contains a 4-cycle that intersects another 4or 5-cyc...
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