Title: On Fan's conjecture about $4$-flow Show Chinese title

Title: 关于Fan关于$4$流的猜想

Abstract: Let $G$ be a bridgeless graph, $C$ is a circuit of $G$. Fan proposed a conjecture that if $G/C$ admits a nowhere-zero 4-flow, then $G$ admits a 4-flow $(D,f)$ such that $E(G)-E(C)\subseteq$ supp$(f)$ and $|\textrm{supp}(f)\cap E(C)|>\frac{3}{4}|E(C)|$. The purpose of this conjecture is to find shorter circuit cover in bridgeless graphs. Fan showed that the conjecture holds for $|E(C)|\le19.$ Wang, Lu and Zhang showed that the conjecture holds for $|E(C)|\le 27$. In this paper, we prove that the conjecture holds for $|E(C)|\le 35.$ Show Chinese abstract

Abstract: 设 $G$ 是一个无桥图， $C$ 是 $G$ 的一个环。 范提出一个猜想，如果$G/C$允许一个非零的4流，则$G$允许一个4流$(D,f)$使得$E(G)-E(C)\subseteq$支持$(f)$并且$|\textrm{supp}(f)\cap E(C)|>\frac{3}{4}|E(C)|$。这个猜想的目的是在无桥图中找到更短的环覆盖。 范证明了该猜想对于$|E(C)|\le19.$成立，王、卢和张证明了该猜想对于$|E(C)|\le 27$成立。在本文中，我们证明了该猜想对于$|E(C)|\le 35.$成立。