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 two vertices $s,t$, find a shortest walk from $s$ to $t$ that visits every vertex. Our main result is that the optimal $5\mathrm{/}4$ coefficient is attained for every terminal pair – including the difficult case where deleting both $s$ and $t$ disconnects the graph. Concretely, every pair of distinct vertices $s,t$ in a simple 2-connected subcubic graph $G$ admits a spanning $s$-$t$ walk of length at most $\lfloor (5n+n_2(G))\mathrm{/}4\rfloor -1$, where $n=\mathrm{\mid }V(G)\mathrm{\mid }$ and $n_2(G)$ is the number of degree-2 vertices; the asymptotic coefficient $5\mathrm{/}4$ cannot be improved, and a simple $O(n^2)$ algorithm finds a walk of length at most $\lfloor (5n+n_2(G))\mathrm{/}4\rfloor$. An edge-rooted even-cover theorem of Wigal, Yoo, and Yu, combined with a short conversion lemma proved here, gives a bound of this form only when $s$ and $t$ are the two endpoints of a given edge; we remove that adjacency restriction. For cubic graphs ($n_2(G)=0$) the bound reads $\lfloor 5n\mathrm{/}4\rfloor -1$, to our knowledge the first $5\mathrm{/}4$ bound for cubic path TSP proved directly rather than through the general path-to-tour reduction.