The edge slide graph of the n-dimensional cube fileThe edge slide graph of Q n Deﬁnition...

The edge slide graph of the n-dimensional cube

Howida AL Fran

Institute of Fundamental SciencesMassey University, Manawatu

8th Australia New Zealand Mathematics ConventionDecember 2014

Howida AL Fran (Massey University) The edge slide graph of Qn ANZMC8 2014 1 / 23

Outline

1 IntroductionCubes and spanning treesEdge movesEdge slides

2 Signatures of spanning trees of Qn

3 Main research goalLocal moves

The n-dimensional cube

DefinitionThe n-dimensional cube is the graph Qnwith

vertices the subsets of the set{1,2, . . . ,n},an edge between two vertices if theydiffer by adding or deleting exactlyone element.

DefinitionLet e be an edge in Qn and let u and v bethe endpoints of e. Then u and v differ byone element i . The direction of e is i .

Spanning trees

DefinitionA spanning tree of a connected graph G is a minimal subset of the edges thatconnects all the vertices.

Edge moves

Definition (Goddard & Swart-1996)

For any spanning tree T of a graph G an edge move is defined as adding oneedge e ∈ G to T and deleting one edge e′ from T so that T + e − e′ is aspanning tree of G.

Tree graph

DefinitionThe 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.

TheoremThe tree graph of a connected graph G is connected.

Example

Edge slides

Definition (Tuffley-2012)

An edge of a spanning tree is slidable if itcan be slid across a 2-dimensional face ofthe cube to give a second spanning tree.

Edge slides

The edge slide graph of Qn

The edge slide graph of Qn is the graphwith

vertices the spanning trees of Qn,an edge between two trees if they arerelated by an edge slide.

The edge slide graph is a subgraph of the tree graph.

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 ordecreases cardinality.

A spanning tree is upright if it has onlydownward edges.

Theorem (Tuffley-2012)

Every spanning tree of Qn is connected toat least one upright tree by a sequence ofedge slides.

An upright tree of Qncorresponds to choosing anelement at each nonemptyvertex.

Signatures of spanning trees of Qn

DefinitionA signature of a spanning tree of Qn isdefined to be (a1,a2, . . . ,an), where ai isthe number of edges in direction i .

The signature of a spanning tree of Qnsatisfies∑n

i=1 ai = 2n − 11 ≤ ai ≤ 2n−1

A spanning tree of Q3 withsignature (2, 3, 2).

Edge slides do not change the signature of a spanning tree.

Characterisation of signatures of spanning trees of Qn

Theorem (Al Fran-2014)

Suppose S = (a1,a2, . . . ,an), where a1 ≥ a2 ≥ · · · ≥ an.Then S is a signature if and only if

∑kj=1 aj ≤ 2n−k (2k − 1), for all k .

ProofUsing Hall’s Marriage Theorem, it suffices to consider upright trees.

For the case S = (3,2,2)

∑kj=1 aj ≤ 2n−k (2k − 1), for all k .

Signatures of spanning trees of Q4

There are 18 signatures of spanning trees of Q4 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)

The edge slide graph of (a1,a2, . . . ,an)

Definition

Let (a1,a2, . . . ,an) be a signature of a spanning tree of Qn. Then the edgeslide graph of (a1,a2, . . . ,an) is the subgraph of the edge slide graph of Qnproduced by trees with signature (a1,a2, . . . ,an).

Edge slides do not change the signature of a spanning tree, so spanning treeswith different signatures will belong to different components.

Main research goal

Conjecture

If ai ≥ 2 for all i , then the edge slide graph of (a1,a2, . . . ,an) is connected.

This conjecture would essentially determine the connected components.Mathematical approach: use “local” moves on the upright trees of Qn todetermine which upright trees are connected.

TheoremConjecture true for Q3 (Henden- 2011).Conjecture true for Q4 (Al Fran- 2014).

Definition (Al Fran-2014)

A local move is a sequence of edge slides that can be applied locally totransform one upright tree of Qn into another.

Main research goal

Conjecture

Main research goal

Conjecture

The V -move

Suppose there is a face F of Qn which islabelled by T as shown in the picture.

Then there exists a sequence of fouredge slides that transforms T into theupright spanning tree T ′, with F labelledas shown in the picture.

The V -move

Suppose there is a face F of Qn which islabelled by T as shown in the picture.Then there exists a sequence of fouredge slides that transforms T into theupright spanning tree T ′, with F labelledas shown in the picture.

The V -move

The path move

Suppose there is a face F of Qn which islabelled by T as shown in the picture.

Then there exists a sequence of fouredge slides that transforms T into theupright spanning tree T ′, with F labelledas shown in the picture.

The path move

The existence of local moves

For all upright trees of Qn with signature (a1,a2, . . . ,an) such that ai ≥ 2, forall i , there is at least one local move.

Assume that the path from{1,2, . . . ,n} to the root is indescending order.Blocking a local move forces thedirection in all vertices ofcardinality 2 containing one.a1 ≥ 2, so there must be at leastone other edge in direction one inthe tree.The local move involving thelowest such one cannot beblocked.

Assume that the path from{1,2, . . . ,n} to the root is indescending order.

Blocking a local move forces thedirection in all vertices ofcardinality 2 containing one.a1 ≥ 2, so there must be at leastone other edge in direction one inthe tree.The local move involving thelowest such one cannot beblocked.

Assume that the path from{1,2, . . . ,n} to the root is indescending order.Blocking a local move forces thedirection in all vertices ofcardinality 2 containing one.

a1 ≥ 2, so there must be at leastone other edge in direction one inthe tree.The local move involving thelowest such one cannot beblocked.

Assume that the path from{1,2, . . . ,n} to the root is indescending order.Blocking a local move forces thedirection in all vertices ofcardinality 2 containing one.a1 ≥ 2, so there must be at leastone other edge in direction one inthe tree.

The local move involving thelowest such one cannot beblocked.

Assume that the path from{1,2, . . . ,n} to the root is indescending order.Blocking a local move forces thedirection in all vertices ofcardinality 2 containing one.a1 ≥ 2, so there must be at leastone other edge in direction one inthe tree.

The local move involving thelowest such one cannot beblocked.

Assume that the path from{1,2, . . . ,n} to the root is indescending order.Blocking a local move forces thedirection in all vertices ofcardinality 2 containing one.a1 ≥ 2, so there must be at leastone other edge in direction one inthe tree.The local move involving thelowest such one cannot beblocked.

The local move graph of Qn

The local move graph of Qn is the graph withvertices the upright trees of Qn,an edge between two trees if they are connected by either the V -move orthe path move.

The local move graph of (a1,a2, . . . ,an), where ai ≥ 2 for all i , is connected⇒the edge slide graph of (a1,a2, . . . ,an) is connected.

The local move of upright trees of signature (2, 2, 3)

The local move graph of upright trees of signature (2,2, 4, 7)

T19 T20

Summary

We characterised the class of n-tuples that are signatures of a spanningtree.We determined the connected components of the edge slide graph of Q4.We proved the existence of a local moves at each upright tree of Qn withsignature (a1,a2, . . . ,an) such that ai ≥ 2, for all i .For future research, we will extend one of the methods that I used for Q4to determine the connected components of the edge slide graph of Qn.

The edge slide graph of the n-dimensional cube fileThe edge slide graph of Q n Deﬁnition...

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