发表机构
School of Mathematical Sciences and LPMC, Nankai University（南开大学数学科学学院和LPMC）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究Tutte的3-流猜想，在允许5-边割时，证明无6至40大小边割的4-边连通图存在无处为零3-流，并推广到本质41-边连通情形。
AI 中文摘要
Tutte 的 $3$-流猜想断言每个 $4$-边连通图都允许一个无处为零的 $3$-流。2013 年，Lovász、Thomassen、Wu 和 Zhang 证明了每个奇 $7$-边连通图都允许一个无处为零的 $3$-流；因此，该猜想对没有大小为 $5$ 的边割的 $4$-边连通图成立。我们考虑允许 $5$-边割的互补情形。我们证明，当允许大小为 $5$ 的边割时，Tutte 的 $3$-流猜想对没有大小从 $6$ 到 $k$ 的任何边割的图成立，其中 $k$ 是一个绝对常数。事实上，$k=40$ 就足够了，并且我们证明了一个更强的版本，其中仅禁止那些大小的非平凡边割。如果删除任意至多 $t-1$ 条边后至多留下一个非平凡分量，则称图为本质 $t$-边连通的。受 Jaeger 的弱 $3$-流猜想的启发，我们证明了关于本质边连通的类似结果：每个 $4$-边连通、本质 $41$-边连通的图都允许一个无处为零的 $3$-流。
英文摘要
Tutte's $3$-flow conjecture asserts that every $4$-edge-connected graph admits a nowhere-zero $3$-flow. In 2013, Lovász, Thomassen, Wu, and Zhang proved that every odd-$7$-edge-connected graph admits a nowhere-zero $3$-flow; consequently, the conjecture holds for $4$-edge-connected graphs with no edge-cut of size $5$. We consider the complementary situation in which $5$-edge-cuts are allowed. We show that, when edge-cuts of size $5$ are permitted, Tutte's $3$-flow conjecture holds for graphs with no edge-cut of any size from $6$ to $k$, where $k$ is an absolute constant. In fact, $k=40$ suffices and we prove a stronger version in which only nontrivial edge-cuts of those sizes are forbidden. A graph is called essentially $t$-edge-connected if deleting any set of at most $t-1$ edges leaves at most one nontrivial component. Motivated by Jaeger's weak $3$-flow conjecture, we prove an analogous result for essential edge connectivity: every $4$-edge-connected, essentially $41$-edge-connected graph admits a nowhere-zero $3$-flow.
