A graph drawn in a surface is a near-quadrangulation if the sum of the lengths of the faces different from 4-faces is bounded by a fixed constant. We leverage duality between colorings and flows to design an efficient algorithm for 3-precoloring-extension in near-quadrangulations of orientable surfaces. Furthermore, we use this duality to strengthen previously known sufficient conditions for 3-colorability of triangle-free graphs drawn in orientable surfaces.

翻译：若曲面上所绘图形中，非四边形的面边长之和受固定常数限制，则该图形称为近四边形分割。我们利用染色与流之间的对偶性，为可定向曲面的近四边形分割中的三预染色扩展问题设计了高效算法。此外，我们借助该对偶性，强化了先前已知的关于可定向曲面上无三角形图可三染色的充分条件。