FLIP GRAPHS, YOKE GRAPHS AND DIAMETER

Transcription

1 FLIP GRAPHS, AND Roy H. Jennings Bar-Ilan University, Israel 9th Sèminaire Lotharingien de Combinatoire 0- September, 0 Bertinoro, Italy FLIP GRAPHS, AND

2 FLIP GRAPH TRIANGULATIONS PERMUTATIONS TREES KNOWN RESULTS FLIP GRAPHS Flip graphs are graphs on sets of objects in which the adjacency relation reflects a minor change in adjacent objects. TYPICAL PROBLEMS Metric properties: distance, diameter, finding antipodes and counting geodesics between them. Algebraic properties: presentations as Cayley/Schreier graphs, automorphism groups, eigenvalues. We generalize known flip graphs into a new family of graphs, namely Yoke graphs. We extend known results, especially the diameter, to this new family. FLIP GRAPHS, AND

3 FLIP GRAPH TRIANGULATIONS PERMUTATIONS TREES KNOWN RESULTS FLIP GRAPHS Flip graphs are graphs on sets of objects in which the adjacency relation reflects a minor change in adjacent objects. TYPICAL PROBLEMS Metric properties: distance, diameter, finding antipodes and counting geodesics between them. Algebraic properties: presentations as Cayley/Schreier graphs, automorphism groups, eigenvalues. We generalize known flip graphs into a new family of graphs, namely Yoke graphs. We extend known results, especially the diameter, to this new family. FLIP GRAPHS, AND

4 FLIP GRAPH TRIANGULATIONS PERMUTATIONS TREES KNOWN RESULTS FLIP GRAPHS Flip graphs are graphs on sets of objects in which the adjacency relation reflects a minor change in adjacent objects. TYPICAL PROBLEMS Metric properties: distance, diameter, finding antipodes and counting geodesics between them. Algebraic properties: presentations as Cayley/Schreier graphs, automorphism groups, eigenvalues. We generalize known flip graphs into a new family of graphs, namely Yoke graphs. We extend known results, especially the diameter, to this new family. FLIP GRAPHS, AND

5 TRIANGULATIONS MOTIVATION AND EXAMPLES FLIP GRAPH TRIANGULATIONS PERMUTATIONS TREES KNOWN RESULTS TRIANGULATION A triangulation of a convex polygon in the plane is its subdivision into triangles using diagonals. FLIP ACTION FLIP GRAPHS, AND

6 TRIANGULATIONS MOTIVATION AND EXAMPLES FLIP GRAPH TRIANGULATIONS PERMUTATIONS TREES KNOWN RESULTS TRIANGULATION A triangulation of a convex polygon in the plane is its subdivision into triangles using diagonals. FLIP ACTION FLIP GRAPHS, AND

7 FLIP GRAPH TRIANGULATIONS PERMUTATIONS TREES KNOWN RESULTS Triangulations of a Hexagon Many variations of the triangulations flip graph have been studied. One such example is the colored flip graph of triangle-free triangulations studied by Adin, Firer and Roichman. FLIP GRAPHS, AND

8 FLIP GRAPH TRIANGULATIONS PERMUTATIONS TREES KNOWN RESULTS COLOURED TRIANGLE FREE TRIANGULATIONS (CTFT) TRIANGLE FREE TRIANGULATION A triangulation is called triangle-free, if it contains no triangle with internal edges (diagonals). FACT Each triangle free triangulation induces two opposite linear orders on its chords. A coloring of a triangulation is a labeling of the chords by one of these orders. FLIP GRAPHS, AND

9 FLIP GRAPH TRIANGULATIONS PERMUTATIONS TREES KNOWN RESULTS COLOURED TRIANGLE FREE TRIANGULATIONS (CTFT) TRIANGLE FREE TRIANGULATION A triangulation is called triangle-free, if it contains no triangle with internal edges (diagonals). FACT Each triangle free triangulation induces two opposite linear orders on its chords. A coloring of a triangulation is a labeling of the chords by one of these orders. FLIP GRAPHS, AND

10 FLIP GRAPH TRIANGULATIONS PERMUTATIONS TREES KNOWN RESULTS ARC PERMUTATIONS ARC PERMUTATION A permutation π S n is called an arc permutation if every prefix (and suffix) in π forms an interval in Z n. EXAMPLE π = is an arc permutation in S. FLIP GRAPHS, AND

11 ARC PERMUTATIONS FLIP GRAPH TRIANGULATIONS PERMUTATIONS TREES KNOWN RESULTS CAYLEY FLIP GRAPHS A (right) Cayley graph X(G, S), where S is a symmetric generating set of G, is an algebraic flip graph in which right multiplication by one of the generators is the flip operation. ARC PERMUTATIONS GRAPH A graph on the arc permutations of S n. Two arc permutations π and σ are connected by an edge if π = σ (i, i + ) for some i n. FLIP GRAPHS, AND

12 ARC PERMUTATIONS FLIP GRAPH TRIANGULATIONS PERMUTATIONS TREES KNOWN RESULTS CAYLEY FLIP GRAPHS A (right) Cayley graph X(G, S), where S is a symmetric generating set of G, is an algebraic flip graph in which right multiplication by one of the generators is the flip operation. ARC PERMUTATIONS GRAPH A graph on the arc permutations of S n. Two arc permutations π and σ are connected by an edge if π = σ (i, i + ) for some i n. FLIP GRAPHS, AND

13 CATERPILLARS MOTIVATION AND EXAMPLES FLIP GRAPH TRIANGULATIONS PERMUTATIONS TREES KNOWN RESULTS CATERPILLAR A (geometric) caterpillar is a non-crossing geometric tree, whose vertices are drawn on a circle, such that the non-leaves form an interval. FLIP ACTION FLIP GRAPHS, AND

14 KNOWN RESULTS MOTIVATION AND EXAMPLES FLIP GRAPH TRIANGULATIONS PERMUTATIONS TREES KNOWN RESULTS Aux. Card. Diameter Proof Arc permutations S n n n n(n ) Similarity with the dominance order on Z n. Caterpillars n-trees n n n(n ) The Hurwitz graph H(S n) on maximal chains in the non crossing partition lattice of S n. CTFT n-gon n n n(n ) The weak order on C n. THEY ARE ALL SCHREIER GRAPHS In all of the cases an affine Weyl group acts transitively on the vertices of the graph. Arc permutations: C n. Caterpillars: C n. Coloured triangle free triangulations: C n. FLIP GRAPHS, AND

15 KNOWN RESULTS MOTIVATION AND EXAMPLES FLIP GRAPH TRIANGULATIONS PERMUTATIONS TREES KNOWN RESULTS Aux. Card. Diameter Proof Arc permutations S n n n n(n ) Similarity with the dominance order on Z n. Caterpillars n-trees n n n(n ) The Hurwitz graph H(S n) on maximal chains in the non crossing partition lattice of S n. CTFT n-gon n n n(n ) The weak order on C n. THEY ARE ALL SCHREIER GRAPHS In all of the cases an affine Weyl group acts transitively on the vertices of the graph. Arc permutations: C n. Caterpillars: C n. Coloured triangle free triangulations: C n. FLIP GRAPHS, AND

16 DEFINITION GENERALIZE THE EXAMPLES YOKE GRAPH Y n,m VERTICES u Y n,m Z n {0, } m Z n m+ i=0 u i 0(modn). ADJACENCY (FLIP) u v if for some 0 i m u j = v j ( j / {i, i + }) and either u i = v i + and u i+ = v i+ (left shift) or u i = v i and u i+ = v i+ + (right shift). EXAMPLE. Y, (00) (00) (00) (000) (000) (00) (00) (00) (00) (00) (000) (000) (00) (000) (000) (00) () (0) () (0) (0) (000) (000) (00000) FLIP GRAPHS, AND

17 DEFINITION GENERALIZE THE EXAMPLES PROPOSITION. CTFT, Arc permutation and Caterpillar graphs are Yoke graphs. BIJECTION FOR ARC PERMUTATIONS The Arc permutations graph A n is isomorphic to Y n,n. ψ : A n Z n {0, } n Z n Example: ψ() = (,, 0, 0,, 0, ): ψ 0 = ψ = ψ = 0 ψ = 0 ψ = ψ = 0 FLIP GRAPHS, AND

18 DEFINITION GENERALIZE THE EXAMPLES PROPOSITION. CTFT, Arc permutation and Caterpillar graphs are Yoke graphs. BIJECTION FOR ARC PERMUTATIONS The Arc permutations graph A n is isomorphic to Y n,n. ψ : A n Z n {0, } n Z n Example: ψ() = (,, 0, 0,, 0, ): ψ 0 = ψ = ψ = 0 ψ = 0 ψ = ψ = 0 FLIP GRAPHS, AND

19 DEFINITION GENERALIZE THE EXAMPLES ψ RESPECTS THE ADJACENCY RELATION Right multiplication by a simple reflection (i, i + ) corresponds to a unit shift between ψ i and ψ i+. For example: ψ() = (,, 0, 0,, 0, ). ψ(()(, )) = (, 0, 0, 0,, 0, ). FOR ALL OF THE EXAMPLES m < n Arc permutations Z n {0, } n Z n (Y n,n ) Caterpillars Z n {0, } n Z n (Y n,n ) CTFT Z n {0, } n Z n (Y n,n ) THEOREM Y n,m is a Schreir graph of the affine Weyl group C m. FLIP GRAPHS, AND

20 DEFINITION GENERALIZE THE EXAMPLES ψ RESPECTS THE ADJACENCY RELATION Right multiplication by a simple reflection (i, i + ) corresponds to a unit shift between ψ i and ψ i+. For example: ψ() = (,, 0, 0,, 0, ). ψ(()(, )) = (, 0, 0, 0,, 0, ). FOR ALL OF THE EXAMPLES m < n Arc permutations Z n {0, } n Z n (Y n,n ) Caterpillars Z n {0, } n Z n (Y n,n ) CTFT Z n {0, } n Z n (Y n,n ) THEOREM Y n,m is a Schreir graph of the affine Weyl group C m. FLIP GRAPHS, AND

21 DEFINITION GENERALIZE THE EXAMPLES ψ RESPECTS THE ADJACENCY RELATION Right multiplication by a simple reflection (i, i + ) corresponds to a unit shift between ψ i and ψ i+. For example: ψ() = (,, 0, 0,, 0, ). ψ(()(, )) = (, 0, 0, 0,, 0, ). FOR ALL OF THE EXAMPLES m < n Arc permutations Z n {0, } n Z n (Y n,n ) Caterpillars Z n {0, } n Z n (Y n,n ) CTFT Z n {0, } n Z n (Y n,n ) THEOREM Y n,m is a Schreir graph of the affine Weyl group C m. FLIP GRAPHS, AND

22 THE OF SKETCH OF PROOF THE OF THEOREM If n m then diam(y n,m ) = n(m+). THEOREM If n m and m n or n m+, then ( m n diam(y n,m ) = d = + ) ( m+n + + ). Otherwise, diam(y n,m ) = d + n m+. FLIP GRAPHS, AND

23 THE OF SKETCH OF PROOF THE OF THEOREM If n m then diam(y n,m ) = n(m+). THEOREM If n m and m n or n m+, then ( m n diam(y n,m ) = d = + ) ( m+n + + ). Otherwise, diam(y n,m ) = d + n m+. FLIP GRAPHS, AND

24 THE OF SKETCH OF PROOF SKETCH OF PROOF THE ECCENTRICITY OF 0 IN Y n,m It can be shown that ecc Yn,m (0) is equal to the value of the diameter in the theorem. Alas, we couldn t prove that 0 is an antipode in Y n,m. DEFINITION (dyoke GRAPHS Z n,m ) Vertices u Z n,m Z n {0, ±} m Z n such that m+ i=0 u i 0(modn). Adjacency (Flip) is the same as in Yoke graphs. THE ECCENTRICITY OF 0 IN Z n,m It can also be shown that ecc Zn,m (0) is equal to the value of the diameter in the theorem. FLIP GRAPHS, AND

25 THE OF SKETCH OF PROOF SKETCH OF PROOF THE ECCENTRICITY OF 0 IN Y n,m It can be shown that ecc Yn,m (0) is equal to the value of the diameter in the theorem. Alas, we couldn t prove that 0 is an antipode in Y n,m. DEFINITION (dyoke GRAPHS Z n,m ) Vertices u Z n,m Z n {0, ±} m Z n such that m+ i=0 u i 0(modn). Adjacency (Flip) is the same as in Yoke graphs. THE ECCENTRICITY OF 0 IN Z n,m It can also be shown that ecc Zn,m (0) is equal to the value of the diameter in the theorem. FLIP GRAPHS, AND

26 THE OF SKETCH OF PROOF SKETCH OF PROOF THE ECCENTRICITY OF 0 IN Y n,m It can be shown that ecc Yn,m (0) is equal to the value of the diameter in the theorem. Alas, we couldn t prove that 0 is an antipode in Y n,m. DEFINITION (dyoke GRAPHS Z n,m ) Vertices u Z n,m Z n {0, ±} m Z n such that m+ i=0 u i 0(modn). Adjacency (Flip) is the same as in Yoke graphs. THE ECCENTRICITY OF 0 IN Z n,m It can also be shown that ecc Zn,m (0) is equal to the value of the diameter in the theorem. FLIP GRAPHS, AND

27 THE OF SKETCH OF PROOF OBSERVATION ϕ u : Y n,m Z n,m v v u is a faithful, injective homomorphism for every u Y n,m. And the images of ϕ u cover Z n,m. LEMMA For every v, u Y n,m, d Yn,m (v, u) = d Zn,m (v u, 0). COROLLARY diam(y n,m ) = ecc Zn,m (0). FLIP GRAPHS, AND

28 THE OF SKETCH OF PROOF OBSERVATION ϕ u : Y n,m Z n,m v v u is a faithful, injective homomorphism for every u Y n,m. And the images of ϕ u cover Z n,m. LEMMA For every v, u Y n,m, d Yn,m (v, u) = d Zn,m (v u, 0). COROLLARY diam(y n,m ) = ecc Zn,m (0). FLIP GRAPHS, AND

29 THE OF SKETCH OF PROOF OBSERVATION ϕ u : Y n,m Z n,m v v u is a faithful, injective homomorphism for every u Y n,m. And the images of ϕ u cover Z n,m. LEMMA For every v, u Y n,m, d Yn,m (v, u) = d Zn,m (v u, 0). COROLLARY diam(y n,m ) = ecc Zn,m (0). FLIP GRAPHS, AND

30 Thank You