Endpoint Conversion for Graphic s-t Path TSP on Cubic Graphs

We study the fixed-endpoint graphic s-t path TSP on simple cubic graphs. Building on the edge-conditioned even-cover theorem of Wigal, Yoo, and Yu for cubic-tour walks, we prove an endpoint-conversion theorem: if G is simple, cubic, and 2-vertex-connected, and if either st is an edge of G or G - {s,t} is connected, then OPT(G,s,t) <= floor(5n/4) - 1. The adjacent-terminal case opens at the terminal edge. The non-adjacent case is reduced to an edge-rooted instance by a constant-size split-and-subdivide gadget that creates a marked edge and contracts back without losing the 5/4 coefficient; the nonseparation hypothesis on {s,t} rules out the rooted theta-chain obstruction. As an application, the terminal condition is automatic on simple cubic 3-edge-connected graphs, and combining the theorem with the trivial lower bound LP >= n - 1 yields an asymptotic path Held-Karp gap upper bound of 5/4. We also prove an adjacent-terminal LP certificate with value exactly n - 1; combined with the Lukotka-Mazak cubic tour lower-bound family, this gives an asymptotic lower bound of 9/8. The coefficient 5/4 is tight for the broader 2-vertex-connected endpoint-conversion theorem, while the exact gap on the 3-edge-connected class remains open in [9/8, 5/4]. An O(n^2)-time algorithm constructs an s-t walk of length at most floor(5n/4).