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(4,2)-choosability of planar graphs with forbidden structures

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 Added by Derrick Stolee
 Publication date 2015
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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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66 - Yangyan Gu , Xuding Zhu 2021
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72 - Mohit Kumbhat , Kevin Moss , 2015
List coloring generalizes graph coloring by requiring the color of a vertex to be selected from a list of colors specific to that vertex. One refinement of list coloring, called choosability with separation, requires that the intersection of adjacent lists is sufficiently small. We introduce a new refinement, called choosability with union separation, where we require that the union of adjacent lists is sufficiently large. For $t geq k$, a $(k,t)$-list assignment is a list assignment $L$ where $|L(v)| geq k$ for all vertices $v$ and $|L(u)cup L(v)| geq t$ for all edges $uv$. A graph is $(k,t)$-choosable if there is a proper coloring for every $(k,t)$-list assignment. We explore this concept through examples of graphs that are not $(k,t)$-choosable, demonstrating sparsity conditions that imply a graph is $(k,t)$-choosable, and proving that all planar graphs are $(3,11)$-choosable and $(4,9)$-choosable.
151 - Ilkyoo Choi , Haihui Zhang 2013
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