Distant 2-Colored Components on Embeddings Part I: Connecting Faces

This is the first in a sequence of three papers in which we prove the following generalization of Thomassen’s 5-choosability theorem: Let $G$ be a finite graph 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 $2^{\mathrm{\Omega }(g)}$ and the precolored components are of distance $2^{\mathrm{\Omega }(g)}$ apart. This provides an affirmative answer to a generalized version of a conjecture of Thomassen and also generalizes a result from 2017 of Dvořák, Lidický, Mohar, and Postle about distant precolored vertices.