A binary clutter is cycling if its packing and covering linear program have integral optimal solutions for all eulerian edge capacities. We prove that the clutter of odd st-walks of a signed graph is cycling if and only if it does not contain as a minor the clutter of odd circuits of K_5 nor the clutter of lines of the Fano matroid. Corollaries of this result include, of many, the characterization for weakly bipartite signed graphs, packing two-commodity paths, packing T-joins with small |T|, a new result on covering odd circuits of a signed graph, as well as a new result on covering odd circuits and odd T-joins of a signed graft.
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机译：如果二进制打包和覆盖线性程序对所有欧拉边缘能力具有整体最优解，那么二进制杂乱就会循环。我们证明，当且仅当它不包含K_5的奇数电路的杂波或Fano拟阵的线的杂波作为辅音时，签名图的奇数st-walk的杂波才会循环。该结果的推论包括，许多特征包括对弱二分符号图的刻画，打包两个商品路径，用小| T |打包T-joins，覆盖符号图的奇数电路的新结果以及新的覆盖有符号移植物的奇数电路和奇数T形接头的结果。
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