1. Rainbow cycles in flip graphs
Torsten Mütze
joint work with Stefan Felsner, Linda Kleist, Leon Sering (TU Berlin)

2. Flip graph of triangulations
Define a graph
Vertices = triangulations of a convex -gon
Edges = flip a single edge of the triangulation
sometimes called associahedron
Example:

3. Properties of the associahedron
What is the diameter of ?
for sufficiently large
[Sleator, Tarjan, Thurston 88]
for , combinatorial proof
[Pournin 14, Adv. Math.]
Realizability as a convex polytope [Ceballos, Santos, Ziegler 15]
Automorphism group, vertex-connectivity, chromatic number [Lee 89], [Hurtado, Noy 99], [Fabila-Monroy et al. 09]
Hamiltonicity [Lucas 87], [Hurtado, Noy 99]
Gray code for triangulations
other Gray codes: - non-crossing perfect matchings [Hernando, Hurtado, Noy 02] - plane spanning trees [Aichholzer et al. 07] - non-crossing partitions, dissections of a convex polygon [Huemer et al. 09]

4. Rainbow cycles
Hamilton cycle: cyclic listing of all combinatorial objects so that each object appears exactly once
Dual problem: cyclic listing of some combinatorial objects so that each flip operation appears exactly once
We call this a rainbow cycle
-rainbow cycle = each edge appears times

5. Motivation binary reflected Gray code [Frank Gray 53]
an algorithm to generate all bitstrings of length by flipping a single bit in each step
balanced Gray code: each bit is flipped many times (requires that is a power of two) [Tootill 53], [Bhat, Savage 96]
this is a rainbow cycle
Our work: first step towards balanced Gray codes for other combinatorial classes (triangulations, matchings etc.)
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6. Rainbow cycles Five settings triangulations of convex polygon
spanning trees on arbitrary point sets
non-crossing matchings
permutations
subsets

7. Our results

8. Triangulations

9. Triangulations

10. Triangulations Theorem: For , the graph has a 2-rainbow cycle. Proof:
Consider the following flip sequence 1 2 3
Claim: is a 2-rainbow cycle
- every edge appears twice:
- uniqueness: can recognize and from intermediate triangulation

11. Matchings

12. Matchings

13. Matchings Theorem: For , the graph has no 1-rainbow cycle.
Assume that is even
Definition: Length of an edge := min. number of points on either side, divided by 2
Centered flip := sum of lengths of quadrilateral equals
centered
not centered
Observation: A flip is centered iff quadrilateral contains origin.

14. Matchings
Lemma: All flips along a rainbow cycle must be centered flips.
Proof:
same number of matching edges of each length
Average length of edges that appear or disappear along a rainbow cycle =
Average length of centered flip =
Average length of non-centered flip <

15. Matchings
Lemma: All flips along a rainbow cycle must be centered flips.
Analyze subgraph of on centered flips to prove non-existence.

16. Subsets Graph : vertices = -element subsets of
edges = subsets that differ in exchanging one element
vertices A B
edge colors
Observation: For , a rainbow cycle is a Hamilton cycle.

17. Subsets
Observation: For , a 1-rainbow cycle can be interpreted as an iterative 2-coloring of the edges of blue = 2-subsets, red = exchanges

18. Subsets

19. Subsets

20. Subsets
Theorem: For odd and , the graph has a 1-rainbow Hamilton cycle.
Proof:
 
 
 
cyclical right-shifts

21. Subsets
Theorem: For odd and , the graph has a rainbow Hamilton cycle. 4 2 1 3 6 7 5
Proof: 3 1 2
starting element 1 2 3 1
 
  2
  3
 
rainbow block
Lemma: Shifting a rainbow block yields a rainbow Hamilton cycle.
Construct rainbow block directly.

22. Open problems
-rainbow cycles for larger values of ? Other combinatorial classes?
Matchings: prove non-existence of 1-rainbow cycle for even number of matching edges
-subsets of : prove existence of 1-rainbow cycle for

23. Thank you!