# David W. Barnette

Refereed publications: Via Math Reviews

Email: barnette@math.ucdavis.edu

Office: 0

Professor David W. Barnette studies graph theory and combinatorial geometry. He has considered various questions about such geometric objects as convex polytopes and triangulations of manifolds. Some of these questions have implications in topology, in particular for invariants such as Gromov norm and decision problems for triangulated manifolds.

One of Professor Barnette's results with a concise statement concerns the
number of k-dimensional faces of an n-dimensional polytope **[1]** **[2]**. He proved
that if such a polytope is simplicial and has v vertices, then it has at
least (d choose k)v - (d+1 choose k+3)(d-1-k) k-dimensional faces. This result
and others convey the general theme that high-dimensional convex polytopes are
necessarily very complicated.

Professor Barnette has also studied the problem of finding minimal
triangulations of surfaces. Here a triangulation must be the "honest" kind in
which two triangles cannot share two vertices without also sharing an edge
connecting them. A triangulation is minimal if there is no way to contract an
edge to a point to obtain a simpler triangulation. He proved that the
projective plane has only two minimal triangulations **[3]**, and later he
and Edelson **[4]** proved that any closed surface has finitely many minimal
triangulations. These results allow for new kinds of arguments by induction on
triangulations of surfaces. They are also similar in spirit to Kuratowski's
classical theorem that a graph is non-planar if and only if it has five
vertices connected by disjoint paths or three vertices connected to three
others by disjoint paths.

### Selected publications

**[1]**The minimum number of vertices of a simple polytope, Israel J. Math. 10 (1971), 121-125.

**[2]** A proof of the lower bound theorem for convex polytopes, Pacific J. Math.
46 (1973), 349-354.

**[3]** Generating the triangulations of the projective plane,
J. Combin. Theory Ser. B 39 (1982), 222-230.

**[4]** All 2-manifolds have finitely many minimal triangulations, Israel J. Math.
67 (1989), 123-128.

**[5]** A construction of 3-connected graphs, Israel J. Math. 86 (1994), 397-407.

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