Grötzsch proved that every triangle-free planar graph is 3-colorable. Thomassen proved that every planar graph of girth at least five is 3-choosable. As for other surfaces, Thomassen proved that there are only finitely many 4-critical graphs of girth at least five embeddable in any fixed surface. This implies a linear-time algorithm for deciding 3-colorablity for graphs of girth at least five on any fixed surface. Dvořák, Král' and Thomas strengthened Thomassen's result by proving that the number of vertices in a 4-critical graph of girth at least five is linear in its genus. They used this result to prove Havel's conjecture that a planar graph whose triangles are pairwise far enough apart is 3-colorable. As for list-coloring, Dvořák proved that a planar graph whose cycles of size at most four are pairwise far enough part is 3-choosable.

In this article, we generalize these results. First we prove a linear isoperimetric bound for 3-list-coloring graphs of girth at least five. Many new results then follow from the theory of hyperbolic families of graphs developed by Postle and Thomas. In particular, it follows that there are only finitely many 4-list-critical graphs of girth at least five on any fixed surface, and that in fact the number of vertices of a 4-list-critical graph is linear in its genus. This provides independent proofs of the above results while generalizing Dvořák's result to graphs on surfaces that have large edge-width and yields a similar result showing that a graph of girth at least five with crossings pairwise far apart is 3-choosable. Finally, we generalize to surfaces Thomassen's result that every planar graph of girth at least five has exponentially many distinct 3-list-colorings. Specifically, we show that every graph of girth at least five that has a 3-list-coloring has $2^{\mathrm{\Omega }(n)-O(g)}$ distinct 3-list-colorings.

Grötzsch证明，每个无三角形的平面图都是3色的。托马森证明，每个周长的平面图至少有5个是3可选的。至于其他表面，Thomassen证明在任何固定表面上都只有有限的许多四临界图，至少有五个可嵌入的周长。这暗示了一种线性时间算法，用于确定周长图在任何固定表面上至少为5的三色性。Dvořák，Král'和Thomas证明了一个四临界周长图中至少五个顶点的线形是线性的，从而加强了Thomassen的结果。他们使用此结果证明了哈维尔的猜想，即三角形成对分布的三角形平面图是3色的。至于列表颜色，

在本文中，我们概括了这些结果。首先，我们证明了围长至少为5的三色列表的线性等距界线。Postle和Thomas提出的双曲图族理论产生了许多新结果。特别地，由此得出结论，在任何固定表面上，至少有五个有限的周长的4个列表临界图，并且实际上，4列表临界图的顶点数在其范畴内是线性的。这提供了上述结果的独立证明，同时将Dvořák的结果推广到具有大边缘宽度的表面上的图形，并且产生了相似的结果，表明周长至少为5且交叉点成对相隔很远的图形是3可选择的。最后，我们泛化到Thomassen' s的结果是，每个围长的平面图至少有5个具有指数不同的3列表着色。具体来说，我们显示出每个围长至少为5且具有3列表颜色的图形都具有$2^{\mathrm{\Omega }（ñ）-Ø（G）}$ 独特的三色列表。