Flips in colorful triangulations
CoRR(2024)
摘要
The associahedron is the graph 𝒢_N that has as nodes all
triangulations of a convex N-gon, and an edge between any two triangulations
that differ in a flip operation, which consists of removing an edge shared by
two triangles and replacing it by the other diagonal of the resulting 4-gon. In
this paper, we consider a large collection of induced subgraphs of
𝒢_N obtained by Ramsey-type colorability properties. Specifically,
coloring the points of the N-gon red and blue alternatingly, we consider only
colorful triangulations, namely triangulations in which every triangle has
points in both colors, i.e., monochromatic triangles are forbidden. The
resulting induced subgraph of 𝒢_N on colorful triangulations is
denoted by ℱ_N. We prove that ℱ_N has a Hamilton cycle
for all N≥ 8, resolving a problem raised by Sagan, i.e., all colorful
triangulations on N points can be listed so that any two cyclically
consecutive triangulations differ in a flip. In fact, we prove that for an
arbitrary fixed coloring pattern of the N points with at least 10 changes of
color, the resulting subgraph of 𝒢_N on colorful triangulations
(for that coloring pattern) admits a Hamilton cycle. We also provide an
efficient algorithm for computing a Hamilton path in ℱ_N that runs
in time 𝒪(1) on average per generated node. This algorithm is based
on a new and algorithmic construction of a tree rotation Gray code for listing
all n-vertex k-ary trees that runs in time 𝒪(k) on average per
generated tree.
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