AI 中文总结
该研究针对次立方图上的图论s-t路径旅行商问题，消除了s、t需为边端点的限制，得到5/4的最优渐近界，还提出了O(n²)算法，为立方图路径旅行商问题提供了首个直接证明的5/4界。
AI 中文摘要
我们研究次立方图（最大度为3）上的图论s-t路径旅行商问题：给定两个顶点s和t，寻找一条从s到t且访问所有顶点的最短游走路径。我们的主要结果是，对于每一对终端顶点，最优系数5/4均成立，其中包括删除s和t都会使图断开的困难情况。具体而言，简单2-连通次立方图G中每一对不同的顶点s、t，都存在一条长度不超过⌊(5n + n₂(G))/4⌋ - 1的生成s-t游走路径，其中n = |V(G)|，n₂(G)是度为2的顶点数量；渐近系数5/4无法进一步改进，且存在简单的O(n²)算法可找到长度不超过⌊(5n + n₂(G))/4⌋的游走路径。Wigal、Yoo和Yu提出的边根偶覆盖定理，结合本文证明的一个简短转换引理，仅在s和t是给定边的两个端点时能给出此类形式的界；我们消除了该邻接限制。对于立方图（n₂(G)=0），该界为⌊5n/4⌋ - 1，据我们所知，这是首个直接证明而非通过通用路径-回路归约得到的立方图路径旅行商问题的5/4界。
英文摘要
We study the graphic $s$-$t$ path TSP on subcubic graphs (maximum degree 3): given distinct vertices $s,t$, find a shortest $s$-$t$ walk that visits every vertex. We prove an upper bound with the asymptotically optimal leading coefficient $5/4$ for every terminal pair, even when $G-\{s,t\}$ is disconnected. Specifically, every simple 2-connected subcubic graph $G$ on $n$ vertices has a spanning $s$-$t$ walk of length at most $\lfloor(5n+n_2(G))/4\rfloor$, where $n_2(G)$ counts its degree-2 vertices. An $O(n^2)$-time algorithm attains this bound. Combining an edge-rooted even-cover theorem of Wigal, Yoo, and Yu (WYY) with an even-cover-to-walk lemma proved here yields this bound for adjacent terminals, a consequence not stated explicitly in their paper. We extend the bound to arbitrary terminal pairs. For cubic graphs, it becomes $\lfloor 5n/4 \rfloor$, to our knowledge the first direct $5/4$ bound for cubic path TSP that does not use the general path-to-tour reduction.
Comments12 pages, 2 figures
