Graphic s-t Path TSP on Cubic Graphs: A 5/4 Bound via Edge-Rooted Excess

We prove a 5/4 endpoint-conversion theorem for graphic s-t path TSP on cubic hosts. If G is simple, cubic, and 2-vertex-connected, and if either st in E (G) or G - s, t is connected, then G has a spanning s-t walk of length at most floor (5n/4) - 1; an O (n²) -time algorithm returns one of length at most floor (5n/4). The proof converts the edge-rooted excess machinery of Wigal, Yoo, and Yu into a path-endpoint statement: adjacent terminals open at the terminal edge, while non-adjacent terminals are handled by a split-and-subdivide gadget that creates a marked edge; together with the nonseparating-terminal condition, the gadget excludes the rooted theta-chain exceptions and contracts back without losing the 5/4 coefficient. This coefficient is asymptotically unavoidable for the theorem: choosing adjacent terminals on the Dvořák-Král'-Mohar tight examples for cubic graphic TSP yields path optimum 5n/4 - O (1). For the LP application, the terminal condition is automatic on simple cubic 3-edge-connected hosts, and the trivial bound LP >= n - 1 gives an asymptotic Held-Karp gap upper bound of 5/4 on this class. We also prove an exact adjacent-terminal LP certificate showing that LP = n - 1, and combine it with the Lukoťka-Mazák 3-connected cubic tour lower-bound family to show that the same asymptotic gap is at least 9/8.