Esperet, de Joannis de Verclos, Le and Thomassé in [SIAM J. Discrete Math., 32(1) (2018), 534–542] introduced the problem that for an odd prime $p$, whether there exists an orientation $D$ of a graph $G$ for any mapping $f:E\left(G\right)\to {\mathbb{Z}}_p^{\ast }$ and any ${\mathbb{Z}}_p$-boundary $b$ of $G$, such that under $D$, at every vertex, the net out $f$-flow is the same as $b\left(v\right)$ in ${\mathbb{Z}}_p$. Such an orientation $D$ is called an $(f,b;p)$-orientation of $G$. Esperet et al. indicated that this problem is closely related to mod $p$-orientations of graphs, including Tutte’s nowhere zero 3-flow conjecture. Utilizing properties of additive bases and contractible configurations, we show that every graph $G$ with Euler genus $g$ and edge-connectivity ${\kappa }^{\prime }\left(G\right)$ admits an $(f,b;p)$-orientation for any mapping $f:E\left(G\right)\to {\mathbb{Z}}_p^{\ast }$ and any ${\mathbb{Z}}_p$-boundary $b$ of $G$, provided ${\kappa }^{\prime }\left(G\right)\ge \left\{\begin{array}{cc}4p-6+\lfloor g∕2\rfloor \hfill & \text{if }g\le 2\text{,}\hfill \\ (p-2)\lfloor \sqrt{6g+0.25}+2.5\rfloor \hfill \\ +1\hfill & \text{if }g\ge 3\text{,}\hfill \\ p\sqrt{4.98g}\hfill & \text{if }g\text{ is sufficiently large.}\hfill \end{array}\right.$

Esperet，de Joannis de Verclos，Le和Thomassé在[SIAM J.Discrete Math。，32（1）（2018），534–542）中引入了一个问题 $p$，是否存在方向 $d$ 图的 $G$ 对于任何映射 $F：Ë（G）\to {\mathbb{ž}}_p^{\ast }$ 和任何 ${\mathbb{ž}}_p$-边界 $b$ 的 $G$，这样 $d$，在每个顶点，净出 $F$-flow与 $b（v）$ 在 ${\mathbb{ž}}_p$。这样的方向$d$ 被称为 $（F，b;p）$的方向 $G$。Esperet等。表示此问题与mod密切相关$p$图的方向，包括Tutte的无处零3流猜想。利用加性碱基和可收缩构型的属性，我们表明每个图$G$ 与欧拉属 $G$ 和边缘连接 ${\kappa }^{\prime }（G）$ 承认 $（F，b;p）$任何映射的方向 $F：Ë（G）\to {\mathbb{ž}}_p^{\ast }$ 和任何 ${\mathbb{ž}}_p$-边界 $b$ 的 $G$，提供 ${\kappa }^{\prime }（G）\ge \left\{\begin{array}{cc}4p-6+\lfloor G∕2\rfloor \hfill & \text{如果 }G\le 2\text{，}\hfill \\ （p-2）\lfloor \sqrt{6G+0。25}+2。5\rfloor \hfill \\ +\mathrm{1个}\hfill & \text{如果 }G\ge 3\text{，}\hfill \\ p\sqrt{4。98G}\hfill & \text{如果 }G\text{ 足够大。}\hfill \end{array}\right.$