Boundary-Preserving Maps between Disks

Suppose that *f: D --> E* is a simplicial map between two disks that sends all edges onto edges. We say that *f* is a *boundary map* if *f* maps the boundary of *D* bijectively to the boundary of *E*. An *even triangulation* of a disk is one with every interior vertex of even degree. If *D* and *E* are even triangulations of the disk, define *D > E* if there is a boundary map from *D* to *E*. This is a partial ordering.

**Conjecture:** *Suppose that* D *is an even triangulation of the disk such that all interior vertices have degree at least 6. If* f: D --> E *is a boundary map, and* E *has all interior vertices of even degree, then* E *and* D *are isomorphic. In other words, show that* D *is a minimal object*.

Since *D* and *E* are even triangulations, they have a 3-coloring. If *D > E*, then the 3-colorings are identical. Let *c* be a 3-coloring of an *n*-cycle, and let *I(c)* be the set of all even disks whose boundary has *c* as a 3-coloring.

**Questions:** *Is there a finite set of minimal elements in* I(c)*? How do you find them?*

For example if the boundary has four vertices, then there are two ways to 3-color the boundary: (1,2,1,2) and (1,2,1,3). Each of these 3-colorings determines a set with exactly one minimal element. If the boundary has five vertices, there is a unique 3-coloring (1,2,1,2,3), and there are two minimal triangulations.

Submitted by: Steve Fisk, Department of Mathematics, Bowdoin College, Brunswick ME 04011-2599 USA.

Send comments to *dan.archdeacon@uvm.edu *and to *fisk@bowdin.edu*

August, 1995