2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/106337The metric polytope m(n) is the polyhedron associated with all semimetrics on n nodes. In 1992 Monique Laurent and Svatopluk Poljak conjectured that every fractional vertex of the metric polytope is adjacent to some integral vertex. The conjecture holds for n<9 and, in particular, for the 1 550 825 600 vertices of m(8). While the overwhelming majority of the known vertices of m(9) satisfy the Laurent-Poljak conjecture, we exhibit a fractional vertex not adjacent to any integral vertex.6 pagesCombinatorics90C27; 52B12text