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

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:

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]

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

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] 000 001 011 010 110 111 101 100 this is a -rainbow cycle Our work: first step towards balanced Gray codes for other combinatorial classes (triangulations, matchings etc.) 224

Rainbow cycles Five settings triangulations of convex polygon spanning trees on arbitrary point sets non-crossing matchings permutations subsets

Our results

Triangulations

Triangulations

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

Matchings

Matchings

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.

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 <

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

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.

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

Subsets

Subsets

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

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 1234567 1     2   3   rainbow block Lemma: Shifting a rainbow block yields a rainbow Hamilton cycle. Construct rainbow block directly.

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

Thank you!