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- W4288096947 abstract "This is the first in a sequence of seven papers in which we prove the following: Suppose we have a graph $G$ embedded on a surface of genus $g$. Then $G$ can be $L$-colored, where $L$ is a list-assignment for $G$ in which every vertex has a 5-list except for a collection of pairwise far-apart components, each precolored with an ordinary 2-coloring, as long as the face-width of $G$ is at leat $2^{Omega(g)}$ and the precolored components are of distance at least $2^{Omega(g)}$ apart. This provides an affirmative answer to a generalized version of a conjecture of Thomassen and also generalizes a result from 2017 of Dvov{r}'ak, Lidick'y, Mohar, and Postle about distant precolored vertices. As an application of this result, we prove (in a follow-up to this sequence of papers) a generalization of a result of Dvov{r}'ak, Lidick'y, and Mohar which states that if a graph drawn in the plane so that all crossings in $G$ are pairwise of distance at least 15 apart, then $G$ is 5-choosable. In our generalization, we prove an analogous result in which planar drawings with pairwise far-apart crossings have been replaced by drawings on arbitrary surfaces with pairwise far-apart matchings with many crossings, where the graph obtained by deleting these matching edges is a high-face-width embedding." @default.
- W4288096947 created "2022-07-28" @default.
- W4288096947 creator A5014021420 @default.
- W4288096947 date "2022-07-21" @default.
- W4288096947 modified "2023-10-17" @default.
- W4288096947 title "Distant 2-Colored Components on Embeddings: Part I" @default.
- W4288096947 doi "https://doi.org/10.48550/arxiv.2207.12531" @default.
- W4288096947 hasPublicationYear "2022" @default.
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