# The edge slide graph of the n-dimensional cube

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1 The edge slide graph of the n-dimensional cube Howida AL Fran Institute of Fundamental Sciences Massey University, Manawatu 8th Australia New Zealand Mathematics Convention December 2014 Howida AL Fran (Massey University) The edge slide graph of Qn ANZMC / 23

2 Outline 1 Introduction Cubes and spanning trees Edge moves Edge slides 2 Signatures of spanning trees of Q n 3 Main research goal Local moves Howida AL Fran (Massey University) The edge slide graph of Qn ANZMC / 23

3 The n-dimensional cube Definition The n-dimensional cube is the graph Q n with vertices the subsets of the set {1, 2,..., n}, an edge between two vertices if they differ by adding or deleting exactly one element. Definition Let e be an edge in Q n and let u and v be the endpoints of e. Then u and v differ by one element i. The direction of e is i. Howida AL Fran (Massey University) The edge slide graph of Qn ANZMC / 23

4 Spanning trees Definition A spanning tree of a connected graph G is a minimal subset of the edges that connects all the vertices. Howida AL Fran (Massey University) The edge slide graph of Qn ANZMC / 23

5 Edge moves Definition (Goddard & Swart-1996) For any spanning tree T of a graph G an edge move is defined as adding one edge e G to T and deleting one edge e from T so that T + e e is a spanning tree of G. Howida AL Fran (Massey University) The edge slide graph of Qn ANZMC / 23

6 Tree graph Definition The tree graph of a connected graph G is the graph with vertices the spanning trees of G, an edge between two trees if they differ by one edge move. Theorem The tree graph of a connected graph G is connected. Howida AL Fran (Massey University) The edge slide graph of Qn ANZMC / 23

7 Example Howida AL Fran (Massey University) The edge slide graph of Qn ANZMC / 23

8 Edge slides Definition (Tuffley-2012) An edge of a spanning tree is slidable if it can be slid across a 2-dimensional face of the cube to give a second spanning tree. Howida AL Fran (Massey University) The edge slide graph of Qn ANZMC / 23

9 Edge slides Definition (Tuffley-2012) An edge of a spanning tree is slidable if it can be slid across a 2-dimensional face of the cube to give a second spanning tree. Howida AL Fran (Massey University) The edge slide graph of Qn ANZMC / 23

10 Edge slides Definition (Tuffley-2012) An edge of a spanning tree is slidable if it can be slid across a 2-dimensional face of the cube to give a second spanning tree. Howida AL Fran (Massey University) The edge slide graph of Qn ANZMC / 23

11 The edge slide graph of Q n Definition (Tuffley-2012) The edge slide graph of Q n is the graph with vertices the spanning trees of Q n, an edge between two trees if they are related by an edge slide. The edge slide graph is a subgraph of the tree graph. Howida AL Fran (Massey University) The edge slide graph of Qn ANZMC / 23

12 Upright spanning trees Root each spanning tree T at. Orient each edge of T towards. An edge is upward or downward depending on whether it increases or decreases cardinality. Definition (Tuffley-2012) A spanning tree is upright if it has only downward edges. Theorem (Tuffley-2012) Every spanning tree of Q n is connected to at least one upright tree by a sequence of edge slides. Howida AL Fran (Massey University) The edge slide graph of Qn ANZMC / 23

13 Upright spanning trees Root each spanning tree T at. Orient each edge of T towards. An edge is upward or downward depending on whether it increases or decreases cardinality. Definition (Tuffley-2012) A spanning tree is upright if it has only downward edges. Theorem (Tuffley-2012) Every spanning tree of Q n is connected to at least one upright tree by a sequence of edge slides. Howida AL Fran (Massey University) The edge slide graph of Qn ANZMC / 23

14 Upright spanning trees Root each spanning tree T at. Orient each edge of T towards. An edge is upward or downward depending on whether it increases or decreases cardinality. Definition (Tuffley-2012) A spanning tree is upright if it has only downward edges. Theorem (Tuffley-2012) Every spanning tree of Q n is connected to at least one upright tree by a sequence of edge slides. Howida AL Fran (Massey University) The edge slide graph of Qn ANZMC / 23

15 Upright spanning trees Root each spanning tree T at. Orient each edge of T towards. An edge is upward or downward depending on whether it increases or decreases cardinality. Definition (Tuffley-2012) A spanning tree is upright if it has only downward edges. Theorem (Tuffley-2012) Every spanning tree of Q n is connected to at least one upright tree by a sequence of edge slides. Howida AL Fran (Massey University) The edge slide graph of Qn ANZMC / 23

16 Upright spanning trees Root each spanning tree T at. Orient each edge of T towards. An edge is upward or downward depending on whether it increases or decreases cardinality. Definition (Tuffley-2012) A spanning tree is upright if it has only downward edges. Theorem (Tuffley-2012) Every spanning tree of Q n is connected to at least one upright tree by a sequence of edge slides. Howida AL Fran (Massey University) The edge slide graph of Qn ANZMC / 23

17 Upright spanning trees Root each spanning tree T at. Orient each edge of T towards. An edge is upward or downward depending on whether it increases or decreases cardinality. Definition (Tuffley-2012) A spanning tree is upright if it has only downward edges. Theorem (Tuffley-2012) Every spanning tree of Q n is connected to at least one upright tree by a sequence of edge slides. Howida AL Fran (Massey University) The edge slide graph of Qn ANZMC / 23

18 Upright spanning trees An upright tree of Q n corresponds to choosing an element at each nonempty vertex. Howida AL Fran (Massey University) The edge slide graph of Qn ANZMC / 23

19 Upright spanning trees An upright tree of Q n corresponds to choosing an element at each nonempty vertex. Howida AL Fran (Massey University) The edge slide graph of Qn ANZMC / 23

20 Upright spanning trees An upright tree of Q n corresponds to choosing an element at each nonempty vertex. Howida AL Fran (Massey University) The edge slide graph of Qn ANZMC / 23

21 Upright spanning trees An upright tree of Q n corresponds to choosing an element at each nonempty vertex. Howida AL Fran (Massey University) The edge slide graph of Qn ANZMC / 23

22 Upright spanning trees An upright tree of Q n corresponds to choosing an element at each nonempty vertex. Howida AL Fran (Massey University) The edge slide graph of Qn ANZMC / 23

23 Upright spanning trees An upright tree of Q n corresponds to choosing an element at each nonempty vertex. Howida AL Fran (Massey University) The edge slide graph of Qn ANZMC / 23

24 Signatures of spanning trees of Q n Definition A signature of a spanning tree of Q n is defined to be (a 1, a 2,..., a n ), where a i is the number of edges in direction i. The signature of a spanning tree of Q n satisfies n i=1 a i = 2 n 1 1 a i 2 n 1 A spanning tree of Q 3 with signature (2, 3, 2). Edge slides do not change the signature of a spanning tree. Howida AL Fran (Massey University) The edge slide graph of Qn ANZMC / 23

25 Characterisation of signatures of spanning trees of Q n Theorem (Al Fran-2014) Suppose S = (a 1, a 2,..., a n ), where a 1 a 2 a n. Then S is a signature if and only if k j=1 a j 2 n k (2 k 1), for all k. Proof Using Hall s Marriage Theorem, it suffices to consider upright trees. Howida AL Fran (Massey University) The edge slide graph of Qn ANZMC / 23

26 Characterisation of signatures of spanning trees of Q n Theorem (Al Fran-2014) Suppose S = (a 1, a 2,..., a n ), where a 1 a 2 a n. Then S is a signature if and only if k j=1 a j 2 n k (2 k 1), for all k. Proof Using Hall s Marriage Theorem, it suffices to consider upright trees. For the case S = (3, 2, 2) Howida AL Fran (Massey University) The edge slide graph of Qn ANZMC / 23

27 Characterisation of signatures of spanning trees of Q n Theorem (Al Fran-2014) Suppose S = (a 1, a 2,..., a n ), where a 1 a 2 a n. Then S is a signature if and only if k j=1 a j 2 n k (2 k 1), for all k. Proof Using Hall s Marriage Theorem, it suffices to consider upright trees. For the case S = (3, 2, 2) Howida AL Fran (Massey University) The edge slide graph of Qn ANZMC / 23

28 Characterisation of signatures of spanning trees of Q n Theorem (Al Fran-2014) Suppose S = (a 1, a 2,..., a n ), where a 1 a 2 a n. Then S is a signature if and only if k j=1 a j 2 n k (2 k 1), for all k. Proof Using Hall s Marriage Theorem, it suffices to consider upright trees. For the case S = (3, 2, 2) Howida AL Fran (Massey University) The edge slide graph of Qn ANZMC / 23

29 Characterisation of signatures of spanning trees of Q n Theorem (Al Fran-2014) Suppose S = (a 1, a 2,..., a n ), where a 1 a 2 a n. Then S is a signature if and only if k j=1 a j 2 n k (2 k 1), for all k. Proof Using Hall s Marriage Theorem, it suffices to consider upright trees. For the case S = (3, 2, 2) Howida AL Fran (Massey University) The edge slide graph of Qn ANZMC / 23

30 Signatures of spanning trees of Q 4 There are 18 signatures of spanning trees of Q 4 up to permutation: (8, 4, 2, 1) (8, 3, 3, 1) (8, 3, 2, 2) (7, 5, 2, 1) (7, 4, 3, 1) (7, 4, 2, 2) (7, 3, 3, 2) (6, 6, 2, 1) (6, 5, 3, 1) (6, 5, 2, 2) (6, 4, 4, 1) (6, 4, 3, 2) (6, 3, 3, 3) (5, 5, 4, 1) (5, 5, 3, 2) (5, 4, 4, 2) (5, 4, 3, 3) (4, 4, 4, 3) Howida AL Fran (Massey University) The edge slide graph of Qn ANZMC / 23

31 The edge slide graph of (a 1, a 2,..., a n ) Definition Let (a 1, a 2,..., a n ) be a signature of a spanning tree of Q n. Then the edge slide graph of (a 1, a 2,..., a n ) is the subgraph of the edge slide graph of Q n produced by trees with signature (a 1, a 2,..., a n ). Edge slides do not change the signature of a spanning tree, so spanning trees with different signatures will belong to different components. Howida AL Fran (Massey University) The edge slide graph of Qn ANZMC / 23

32 Main research goal Conjecture If a i 2 for all i, then the edge slide graph of (a 1, a 2,..., a n ) is connected. This conjecture would essentially determine the connected components. Mathematical approach: use local moves on the upright trees of Q n to determine which upright trees are connected. Theorem Conjecture true for Q 3 (Henden- 2011). Conjecture true for Q 4 (Al Fran- 2014). Definition (Al Fran-2014) A local move is a sequence of edge slides that can be applied locally to transform one upright tree of Q n into another. Howida AL Fran (Massey University) The edge slide graph of Qn ANZMC / 23

33 Main research goal Conjecture If a i 2 for all i, then the edge slide graph of (a 1, a 2,..., a n ) is connected. This conjecture would essentially determine the connected components. Mathematical approach: use local moves on the upright trees of Q n to determine which upright trees are connected. Theorem Conjecture true for Q 3 (Henden- 2011). Conjecture true for Q 4 (Al Fran- 2014). Definition (Al Fran-2014) A local move is a sequence of edge slides that can be applied locally to transform one upright tree of Q n into another. Howida AL Fran (Massey University) The edge slide graph of Qn ANZMC / 23

34 Main research goal Conjecture If a i 2 for all i, then the edge slide graph of (a 1, a 2,..., a n ) is connected. This conjecture would essentially determine the connected components. Mathematical approach: use local moves on the upright trees of Q n to determine which upright trees are connected. Theorem Conjecture true for Q 3 (Henden- 2011). Conjecture true for Q 4 (Al Fran- 2014). Definition (Al Fran-2014) A local move is a sequence of edge slides that can be applied locally to transform one upright tree of Q n into another. Howida AL Fran (Massey University) The edge slide graph of Qn ANZMC / 23

35 The V -move Theorem (Al Fran-2014) Suppose there is a face F of Q n which is labelled by T as shown in the picture. Howida AL Fran (Massey University) The edge slide graph of Qn ANZMC / 23

36 The V -move Theorem (Al Fran-2014) Suppose there is a face F of Q n which is labelled by T as shown in the picture. Then there exists a sequence of four edge slides that transforms T into the upright spanning tree T, with F labelled as shown in the picture. Howida AL Fran (Massey University) The edge slide graph of Qn ANZMC / 23

37 The V -move Theorem (Al Fran-2014) Suppose there is a face F of Q n which is labelled by T as shown in the picture. Then there exists a sequence of four edge slides that transforms T into the upright spanning tree T, with F labelled as shown in the picture. Howida AL Fran (Massey University) The edge slide graph of Qn ANZMC / 23

38 The path move Theorem (Al Fran-2014) Suppose there is a face F of Q n which is labelled by T as shown in the picture. Howida AL Fran (Massey University) The edge slide graph of Qn ANZMC / 23

39 The path move Theorem (Al Fran-2014) Suppose there is a face F of Q n which is labelled by T as shown in the picture. Then there exists a sequence of four edge slides that transforms T into the upright spanning tree T, with F labelled as shown in the picture. Howida AL Fran (Massey University) The edge slide graph of Qn ANZMC / 23

40 The path move Theorem (Al Fran-2014) Suppose there is a face F of Q n which is labelled by T as shown in the picture. Then there exists a sequence of four edge slides that transforms T into the upright spanning tree T, with F labelled as shown in the picture. Howida AL Fran (Massey University) The edge slide graph of Qn ANZMC / 23

41 The existence of local moves Theorem (Al Fran-2014) For all upright trees of Q n with signature (a 1, a 2,..., a n ) such that a i 2, for all i, there is at least one local move. Howida AL Fran (Massey University) The edge slide graph of Qn ANZMC / 23

42 The existence of local moves Theorem (Al Fran-2014) For all upright trees of Q n with signature (a 1, a 2,..., a n ) such that a i 2, for all i, there is at least one local move. Assume that the path from {1, 2,..., n} to the root is in descending order. Howida AL Fran (Massey University) The edge slide graph of Qn ANZMC / 23

43 The existence of local moves Theorem (Al Fran-2014) For all upright trees of Q n with signature (a 1, a 2,..., a n ) such that a i 2, for all i, there is at least one local move. Assume that the path from {1, 2,..., n} to the root is in descending order. Blocking a local move forces the direction in all vertices of cardinality 2 containing one. Howida AL Fran (Massey University) The edge slide graph of Qn ANZMC / 23

44 The existence of local moves Theorem (Al Fran-2014) For all upright trees of Q n with signature (a 1, a 2,..., a n ) such that a i 2, for all i, there is at least one local move. Assume that the path from {1, 2,..., n} to the root is in descending order. Blocking a local move forces the direction in all vertices of cardinality 2 containing one. Howida AL Fran (Massey University) The edge slide graph of Qn ANZMC / 23

45 The existence of local moves Theorem (Al Fran-2014) For all upright trees of Q n with signature (a 1, a 2,..., a n ) such that a i 2, for all i, there is at least one local move. Assume that the path from {1, 2,..., n} to the root is in descending order. Blocking a local move forces the direction in all vertices of cardinality 2 containing one. Howida AL Fran (Massey University) The edge slide graph of Qn ANZMC / 23

46 The existence of local moves Theorem (Al Fran-2014) For all upright trees of Q n with signature (a 1, a 2,..., a n ) such that a i 2, for all i, there is at least one local move. Assume that the path from {1, 2,..., n} to the root is in descending order. Blocking a local move forces the direction in all vertices of cardinality 2 containing one. a 1 2, so there must be at least one other edge in direction one in the tree. Howida AL Fran (Massey University) The edge slide graph of Qn ANZMC / 23

47 The existence of local moves Theorem (Al Fran-2014) For all upright trees of Q n with signature (a 1, a 2,..., a n ) such that a i 2, for all i, there is at least one local move. Assume that the path from {1, 2,..., n} to the root is in descending order. Blocking a local move forces the direction in all vertices of cardinality 2 containing one. a 1 2, so there must be at least one other edge in direction one in the tree. Howida AL Fran (Massey University) The edge slide graph of Qn ANZMC / 23

48 The existence of local moves Theorem (Al Fran-2014) For all upright trees of Q n with signature (a 1, a 2,..., a n ) such that a i 2, for all i, there is at least one local move. Assume that the path from {1, 2,..., n} to the root is in descending order. Blocking a local move forces the direction in all vertices of cardinality 2 containing one. a 1 2, so there must be at least one other edge in direction one in the tree. The local move involving the lowest such one cannot be blocked. Howida AL Fran (Massey University) The edge slide graph of Qn ANZMC / 23

49 The local move graph of Q n Definition (Al Fran-2014) The local move graph of Q n is the graph with vertices the upright trees of Q n, an edge between two trees if they are connected by either the V -move or the path move. The local move graph of (a 1, a 2,..., a n ), where a i 2 for all i, is connected the edge slide graph of (a 1, a 2,..., a n ) is connected. Howida AL Fran (Massey University) The edge slide graph of Qn ANZMC / 23

50 The local move of upright trees of signature (2, 2, 3) Howida AL Fran (Massey University) The edge slide graph of Qn ANZMC / 23

51 The local move graph of upright trees of signature (2, 2, 4, 7) T5 T6 T4 T3 T2 T1 T13 T12 T7 T10 T19 T20 T11 T8 T21 T22 T14 T9 T24 T23 T18 T17 T16 T15 Howida AL Fran (Massey University) The edge slide graph of Qn ANZMC / 23

52 Summary We characterised the class of n-tuples that are signatures of a spanning tree. We determined the connected components of the edge slide graph of Q 4. We proved the existence of a local moves at each upright tree of Q n with signature (a 1, a 2,..., a n ) such that a i 2, for all i. For future research, we will extend one of the methods that I used for Q 4 to determine the connected components of the edge slide graph of Q n. Howida AL Fran (Massey University) The edge slide graph of Qn ANZMC / 23

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