This note presents several results in graph theory inspired by the author's work in the proof theory of linear logic; these results are purely combinatorial and do not involve logic. We show that trails avoiding forbidden transitions, properly arc-colored directed trails and rainbow paths for complete multipartite color classes can be found in linear time, whereas finding rainbow paths is NP-complete for any other restriction on color classes. For the tractable cases, we also state new structural properties equivalent to Kotzig's theorem on the existence of bridges in unique perfect matchings. Another result on graphs equipped with unique perfect matchings that we prove here is the combinatorial counterpart of a theorem due to Bellin in linear logic: a connection between blossoms and bridge deletion orders.

中文翻译：

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非循环性的约束寻路和结构

本笔记介绍了受作者在线性逻辑证明理论中的工作启发的图论中的几个结果；这些结果纯粹是组合的，不涉及逻辑。我们表明，可以在线性时间内找到避免禁止转换的路径、正确的弧形定向路径和完整的多部分颜色类别的彩虹路径，而找到彩虹路径对于颜色类别的任何其他限制都是 NP 完全的。对于易处理的情况，我们还陈述了新的结构性质，等价于 Kotzig 定理关于桥存在唯一完美匹配的定理。我们在此证明的配备独特完美匹配的图的另一个结果是线性逻辑中贝林定理的组合对应：开花和桥删除顺序之间的连接。