Let $G$ be a graph embedded on a surface $S_\varepsilon$ with Euler genus$\varepsilon > 0$, and let $P\subseteq V(G)$ be a set of vertices mutually atdistance at least 4 apart. Suppose all vertices of $G$ have$H(\varepsilon)$-lists and the vertices of $P$ are precolored, where$H(\varepsilon)=\Big\lfloor\frac{7 + \sqrt{24\varepsilon + 1}}{2}\Big\rfloor$is the Heawood number. We show that the coloring of $P$ extends to alist-coloring of $G$ and that the distance bound of 4 is best possible. Ourresult provides an answer to an analogous question of Albertson about extendinga precoloring of a set of mutually distant vertices in a planar graph to a5-list-coloring of the graph and generalizes a result of Albertson andHutchinson to list-coloring extensions on surfaces.
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机译：假设$ G $是嵌入到表面$ S_ \ varepsilon $中且Euler属$ \ varepsilon> 0 $的图，并且让$ P \ subseteq V（G）$是彼此间隔至少4个的一组顶点。假设$ G $的所有顶点都具有$ H（\ varepsilon）$-lists，并且$ P $的顶点已预着色，其中$ H（\ varepsilon）= \ Big \ lfloor \ frac {7 + \ sqrt {24 \ varepsilon + 1}} {2} \ Big \ rfloor $是Heawood号码。我们表明，$ P $的着色扩展到$ G $的列表着色，并且距离边界4最好。我们的结果为阿尔伯森的一个类似问题提供了答案，该问题涉及将平面图中一组相互远离的顶点的预着色扩展到图的5列表着色，并将阿尔伯森和哈钦森的结果推广到表面上扩展的列表着色。
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