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تسجيل مستخدم جديد(4,2)-choosability of planar graphs with forbidden structures
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All planar graphs are 4-colorable and 5-choosable, while some planar graphs are not 4-choosable. Determining which properties guarantee that a planar graph can be colored using lists of size four has received significant attention. In terms of constraining the structure of the graph, for any $ell in {3,4,5,6,7}$, a planar graph is 4-choosable if it is $ell$-cycle-free. In terms of constraining the list assignment, one refinement of $k$-choosability is choosability with separation. A graph is $(k,s)$-choosable if the graph is colorable from lists of size $k$ where adjacent vertices have at most $s$ common colors in their lists. Every planar graph is $(4,1)$-choosable, but there exist planar graphs that are not $(4,3)$-choosable. It is an open question whether planar graphs are always $(4,2)$-choosable. A chorded $ell$-cycle is an $ell$-cycle with one additional edge. We demonstrate for each $ell in {5,6,7}$ that a planar graph is $(4,2)$-choosable if it does not contain chorded $ell$-cycles.
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