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.

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August, 1995