图谜II.1：贾因第二单位向量流猜想的反例
Graph Puzzles II.1: Counterexamples to Jain's Second Unit Vector Flows Conjecture
AI总结：
本文通过构造反例，证明贾因关于单位向量流的第二个猜想不成立，展示了需要额外值{-5,5}的点集，挑战了流存在性的假设。
摘要：
在图G上，3维无零流是指每条边赋以单位范数的3维向量（对应2维单位球S²的点）。K.贾因提出了两个相关猜想。第一个猜想认为所有桥连图都存在这样的流。第二个猜想则要求将S²的点赋值{-4,-3,-2,-1,1,2,3,4}，使得反点获得相反值，且任意大圆上的三点等距点值和为零。若两者成立，则共同推导出图论中的图流猜想。本文通过构造反例，证明第二个猜想不成立，展示了需要额外值{-5,5}的点集。
英文摘要：
A $3$-dimensional nowhere-zero flow on a graph $G$ is a flow where each edge is assigned a $3$-dimensional vector with unit norm (which corresponds to the points of a $2$-dimensional unit sphere $S^2$). K. Jain posed two conjectures related to this idea. First one suggests that such a flow exists for all bridgeless graphs. The second conjecture states that we can assign values $\{-4,-3,-2,-1,1,2,3,4\}$ to the points of $S^2$, such that antipodal points get opposite values, and values of any three equidistant points on great circles sum to zero. If both conjectures would be true, together they would imply Tutte's 5-flow conjecture. We show 2 counterexamples to the second conjecture, by constructing sets of points each of which additionally requires values $\{-5, 5\}$. Github: https://github.com/gexahedron/unit-vector-flows
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
