A 5/4 bound for graphic $s$-$t$ path TSP on subcubic graphs
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.