A counterexample to a conjecture of Laurent and Poljak

Details A counterexample to a conjecture of Laurent and Poljak A counterexample to a conjecture of Laurent and Poljak

2005-12-21

arxiv.org/abs/math/0512493v1

Mathematics

Combinatorics

6 pages

Scientific paper

The 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.

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