CARÁCTER

In this section we define the local move graph of Qn using the V-move and the path move that we introduced in this chapter. Then we define the local move graph of a signature of Q_n.

Definition 4.3.1. The local move graph of Qn is the graph L(Qn) whose vertices are the upright spanning trees, with an edge between two vertices (upright spanning trees) if they are related by either the V-move or the path move.

Definition 4.3.2. Let S = (a₁, . . . , an) be a signature of a spanning tree of Qn. Then the

local move graph of signature S, denoted L(S), is defined to be the subgraph of L(Q_n)

produced by the trees with signature (a₁, . . . , an). Note thatL(a1, . . . , an)is always a union of

components of L(Qn).

4.3.1

The local move graph of the irreducible signature (2, 2, 3)

In this section we count the number of upright spanning trees of Q₃ with signature (2, 2, 3), and then we show that the local move graph of signature (2, 2, 3) is connected. Before working with signature (2, 2, 3) we establish the following lemma.

Lemma 4.3.3. Let U_i(1) = (0, . . . ,1

ith

,0, . . . ,0), where i ∈ [n]. Then the number of upright spanning forests of Q_n with signature U_i(1) is 2n−1_.

Proof. An upright spanning forest ofQ_n with signature U_i(1) consists of 2n−2trivial trees and a rooted upright tree with a single edge in direction i. Sincei can be taken from 2n−1 _vertices

of Qn, we have 2n−1 _{upright rooted spanning forests ofQn with signature U}(1)

i .

Observation 4.3.4. There are n2n−1 _{upright spanning forests of Q}

n with signature

U(1)

1 = (1,0, . . . ,0) up to permutation.

Proof. An upright spanning tree ofQ₃ with signature (2, 2, 3) consists of an upright spanning forest of Q₂ in F₃3+ with signature (1, 0) or signature (0, 1), and an upright spanning tree of

Q₂ in F₃3− with signature (1, 2) or signature (2, 1) respectively. The positions of the 3-edges are forced by the choice of edge in F₃3+. There are two upright spanning forests of Q₂ with signature (1, 0) (by Lemma 4.3.3) and one upright spanning tree of Q₂ with signature (1, 2) up to permutation. Thus, altogether there are a total of 2(1)+2(1)= 4 upright spanning trees of Q₃ with signature (2, 2, 3).

Note that the above lemma can also be proved by observing that the monomial q2 1q22q33

appears with coefficient 4 in the generating function

q₁q₂q₃(q₁+q₂)(q₁+q₃)(q₂+q₃)(q₁+q₂ +q₃)

for sections of P_≥3₁.

Lemma 4.3.6. The edge slide graph of the irreducible signature (2, 2, 3) is connected.

Proof. Since every spanning tree of Q₃ is connected to at least one upright spanning tree by a series of edge slides (Tuffley [11]), it suffices to show that the local move graph of the signature (2, 3, 3) is connected. We label the trees as discussed in Section 2.2.5.2, and show that any upright spanning tree of Q₃ with the irreducible signature (2, 2, 3) can be transformed into the upright spanning tree 1233 (Figure 4.8(iv) on page 46) by a local move.

Consider the tree 3213, shown in Figure 4.8(i). Applying theV-move on the faceF₃1+(green edges), the labels 1 and 3 are swapped. Therefore the tree 3213 is transformed to the tree 1233 (Figure 4.8(iv)). Note that the tree 3213 can also be transformed to the tree 2133 (Fig- ure 4.8(iii)) by applying the path move on the same F₃1+.

Consider the tree 3132, shown in Figure 4.8(ii). Applying the path move on the face F₃2+ (green edges), label 2 moves to {1,2}, label 1 moves to {1,2,3} and label 3 moves to {2,3}. Therefore the tree 3132 is transformed to the tree 1233 (Figure 4.8(iv)). Note that the tree 3132 can also be transformed to the tree 2133 (Figure 4.8(iii)) by applying theV-move onF₃2+. Consider the tree 2133, shown in Figure 4.8(iii). Applying the V-move on the face F₃1+ (green edges), the labels 1 and 2 are swapped. Therefore the tree 2133 is transformed to the tree 1233 (Figure 4.8(iv)).

By the definition of the local move graph L(2,2,3), there is a local move from each upright spanning tree 3213, 3132 and 2133 to the upright spanning tree 1233, hence all four upright spanning trees are in the same connected component of L(2,2,3). Therefore the local move graph L(2,2,3) is connected.

Since every spanning tree ofQ₃ is connected to an upright spanning tree, spanning trees of

Q₃ with signature (2, 2, 3) lie in a single component of the edge slide graph ofQ₃ and therefore

E(2,2,3) is connected.

Figure 4.9 on page 47 shows the local move graph of the irreducible signature (2, 2, 3).

4.4

Summary map

As stated in this chapter, upright spanning trees with the same signature which belong to the same connected component of the local move graph of Qn, also belong to the same connected component of the edge slide graph of Qn. In the following chapters we use local moves, the

1∈ {1,2} 2∈ {2,3} 3∈ {1,2,3} ∅ {1} 3∈ {1,3} {2} {3} 2∈ {1,2} 3∈ {2,3} 3∈ {1,2,3} ∅ {1} 1∈ {1,3} {2} {3} (ii) (i) 2∈ {1,2} 3∈ {2,3} 1∈ {1,2,3} ∅ {1} 3∈ {1,3} {2} {3} 1∈ {1,2} 3∈ {2,3} 2∈ {1,2,3} ∅ {1} 3∈ {1,3} {2} {3} (iv) (iii)

Figure 4.8: The four upright spanning trees (bold edges) of Q₃ with the irreducible signature (2, 2, 3), where the possible V-moves and path moves (green) are shown. The labels are (i) 3213, (ii) 3132, (iii) 2133, (iv) 1233.

Figure 4.9: The local move graph of signature (2, 2, 3), with the V-move (blue) and the path move (red).

Splitting spanning trees and signatures of

Q

_n