Abstract
The Four Color Theorem states that every planar graph is properly 4-colorable. Moreover, it is well known that there are planar graphs that are non-$4$-list colorable. In this paper we investigate a problem combining proper colorings and list colorings. We ask whether the vertex set of every planar graph can be partitioned into two subsets where one subset induces a bipartite graph and the other subset induces a $2$-list colorable graph. We answer this question in the negative strengthening the result on non-$4$-list colorable planar graphs.
Highlights
In this note we consider simple planar graphs
Since 1993 it is known by Thomassen [6] and Voigt [7] that every planar graph is 5-list colorable but there are planar graphs that are non-4-list colorable
Using the Four Color Theorem [1, 5], it is easy to see that every planar graph is L-colorable for every 3-common 4-assignment L
Summary
In this note we consider simple planar graphs. Since 1993 it is known by Thomassen [6] and Voigt [7] that every planar graph is 5-list colorable but there are planar graphs that are non-4-list colorable. If G is L-colorable for all possible k-assignments G is called k-list colorable. Using the Four Color Theorem [1, 5], it is easy to see that every planar graph is L-colorable for every 3-common 4-assignment L.
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