Gallai's conjecture with two exceptional even vertices

Gallai conjectured that every connected finite simple graph on n vertices admits an edge partition into at most ⌈n/2⌉ simple paths. Botler and Sambinelli proved this when every even-degree vertex has at most three even neighbours. Fan, Hou and Zhou exempted one designated even vertex of positive degree and ensured that at least two paths ended there. Xie obtained the same endpoint conclusion for either of two adjacent designated even vertices, in separate partitions, when every other even vertex has at most three even neighbours. We prove that the same endpoint conclusion holds without requiring the two designated even vertices to be adjacent. Let F be the subgraph induced by the even-degree vertices of a connected finite simple graph G on n vertices, and let h and x be distinct vertices of F. Suppose every other vertex of F has degree at most three in F. Then for each of h and x there is an edge partition into at most ⌈n/2⌉ simple paths, at least two of which end at that vertex. Consequently, Gallai's bound holds whenever at most two vertices of F have degree greater than three. The two choices may use different partitions. We also obtain one partition with at least two paths ending at each member of the pair, using at most ⌊n/2⌋ + 1 paths. At odd order this is Gallai's bound. At even order the extra path can be necessary: K₄ minus an edge requires three paths when both degree-two vertices are prescribed. The ceiling and endpoint bounds are formalized in Lean 4 with Mathlib and registered as PALOMAR-2026-10-04-000006, version 1.