图$G=(V (G), E (G))$上的一个处处非零$k$-流是指如下定义的一个对$(D, f)$, 其中$D$是边集$E (G)$上的一个定向, $f\colon E (G)\to\{\pm1, \pm2, \cdots, \pm (k-1)\}$是边集上的函数, 且满足每个顶点的总流出与总流入相等。这一概念由Tutte引入, 作为面着色的扩展。Tutte在1954年提出了著名的5-流猜想: 每个无桥图都存在处处非零的5-流。尽管该猜想已在一些图类中得到验证, 但至今仍未完全解决。本文证明了每个欧拉亏格至多为20的无桥图都存在一个处处非零的5-流, 从而改进了若干已知结果。

A nowhere-zero $k$-flow on a graph $G=(V(G), E(G))$ is a pair $(D, f)$, where $D$ is an orientation on $E(G)$ and $f\colon E(G)\to \{\pm1, \pm2, \cdots, \pm(k-1)\}$ is a function such that the total outflow equals to the total inflow at each vertex. This concept was introduced by Tutte as an extension of face colorings, and Tutte in 1954 conjectured that every bridgeless graph admits a nowhere-zero 5-flow, known as the 5-Flow Conjecture. This conjecture is verified for some graph classes and remains unresolved as of today. In this paper, we show that every bridgeless graph of Euler genus at most 20 admits a nowhere-zero 5-flow, which improves several known results.