Journal of Algebraic Combinatorics

, Volume 41, Issue 2, pp 385–395 | Cite as

Lit-only sigma-game on nondegenerate graphs

Open Access
Article

Abstract

A configuration of the lit-only \(\sigma \)-game on a graph \(\Gamma \) is an assignment of one of two states, on or off, to each vertex of \(\Gamma \). Given a configuration, a move of the lit-only \(\sigma \)-game on \(\Gamma \) allows the player to choose an on vertex \(s\) of \(\Gamma \) and change the states of all neighbors of \(s\). Given an integer \(k\), the underlying graph \(\Gamma \) is said to be \(k\)-lit if for any configuration, the number of on vertices can be reduced to at most \(k\) by a finite sequence of moves. We give a description of the orbits of the lit-only \(\sigma \)-game on nondegenerate graphs \(\Gamma \) which are not line graphs. We show that these graphs \(\Gamma \) are 2-lit and provide a linear algebraic criterion for \(\Gamma \) to be 1-lit.

Keywords

Group action Lit-only \(\sigma \)-game Nondegenerate graph 

Mathematics Subject Classification

Primary 05C57 Secondary 15A63 20F55 

1 Introduction

The notion of the \(\sigma \)-\(game\) on finite graphs \(\Gamma \) was first introduced by Sutner [17, 18] around 1989. A configuration of the \(\sigma \)-game on \(\Gamma \) is an assignment of one of two states, on or off, to each vertex of \(\Gamma \). Given a configuration, a move consists of choosing a vertex of \(\Gamma \), followed by changing the states of all of its neighbors. If only on vertices can be chosen in each move, we come to the variation: lit-only\(\sigma \)-game. Starting from an initial configuration, the goal of the lit-only \(\sigma \)-game on \(\Gamma \) is to minimize the number of on vertices of \(\Gamma \), or to reach an assigned configuration by a finite sequence of moves.

Given an integer \(k\), the underlying graph \(\Gamma \) is said to be \(k\)-\(lit\) if for any configuration, the number of on vertices can be reduced to at most \(k\) by a finite sequence of moves. More precisely, we are interested in the orbits of the lit-only \(\sigma \)-game on \(\Gamma \) and the smallest integer \(k\), the minimum light number of \(\Gamma \) [19], for which \(\Gamma \) is \(k\)-lit. The notion of lit-only \(\sigma \)-games occurred implicitly in the study of equivalence classes of Vogan diagrams. The Borel-de Siebenthal theorem [2] showed that every Vogan diagram is equivalent to one with a single-painted vertex, which implies that each simply-laced Dynkin diagram is \(1\)-lit. The equivalence classes of Vogan diagrams were described by Chuah and Hu [7]. A conjecture made by Chang [5, 6] that any tree with \(k\) leaves is \(\lceil k/2 \rceil \)-lit was confirmed by Wang and Wu [19], where the name “lit-only \(\sigma \)-game” was coined.

The lit-only \(\sigma \)-game on a simple graph \(\Gamma \) is simply the natural action of a certain subgroup \(H_\Gamma \) of the general linear group over \(\mathbb {F}_2\) [19]. Under the assumption that \(\Gamma \) is the line graph of a simple graph \(G\), Wu [21] described the orbits of the lit-only \(\sigma \)-game on \(\Gamma \) and gave a characterization for the minimum light number of \(\Gamma \). Moreover, if \(G\) is a tree of order \(n\ge 3 \), Wu showed that \(H_\Gamma \) is isomorphic to the symmetric group on \(n\) letters. Weng and the author [13] determined the structure of \(H_\Gamma \) without any assumption on \(G\). The lit-only \(\sigma \)-game on a simple graph \(\Gamma \) can also be considered as a representation \(\kappa _{\Gamma }\) of the simply-laced Coxeter group \(W_{\Gamma }\) over \(\mathbb {F}_2\) [12]. The dual representation of \(\kappa _{\Gamma }\) preserves a certain symplectic form \(B_{\Gamma }\). The two representations are equivalent whenever the form \(B_{\Gamma }\) is nondegenerate. From this viewpoint it is natural to partition simple connected graphs into two classes according as \(B_{\Gamma }\) is degenerate or nondegenerate.

In this paper, we treat nondegenerate graphs \(\Gamma \) which are not line graphs. We show that \(H_{\Gamma }\) is isomorphic to an orthogonal group, followed by a description of the orbits of lit-only \(\sigma \)-game on \(\Gamma \) (Theorem 3.1). Moreover, we show that these graphs \(\Gamma \) are 2-lit and provide a linear algebraic criterion for \(\Gamma \) to be 1-lit (Theorem 3.2). Combining Theorem 3.1, Theorem 3.2, and those in [13, 21], the study of the lit-only \(\sigma \)-game on nondegenerate graphs is quite completed, and the focus for further research is on degenerate graphs.

2 Preliminaries

From now on, let \(\Gamma =(S,R)\) denote a finite simple connected graph with vertex set \(S\) and edge set \(R\). Let \(\mathbb {F}_2\) denote the two-element field \(\{0,1\}\). Let \(V\) denote an \(\mathbb {F}_2\)-vector space that has a basis \(\{\alpha _s~|~s\in S\}\) in one-to-one correspondence with \(S\). Let \(V^*\) denote the dual space of \(V\). For each \(s\in S\), we define \(f_s\in V^*\) by
$$\begin{aligned} f_s(\alpha _t)=\left\{ \begin{array}{ll} 1 &{}\quad \mathrm{if}\,\,\, s=t,\\ 0 &{}\quad \mathrm{else} \end{array} \right. \end{aligned}$$
(1)
for all \(t\in S\). The set \(\{f_s~|~s\in S\}\) forms a basis of \(V^*\) and is called the basis of\(V^*\)dual to\(\{\alpha _s~|~s\in S\}\). Each configuration \(f\) of the lit-only \(\sigma \)-game on \(\Gamma \) is interpreted as the vector
$$\begin{aligned} \sum _{\mathrm{on\,\, vertices}\,\, s} f_s\in V^*. \end{aligned}$$
(2)
If all vertices of \(\Gamma \) are assigned the off state by \(f\), we interpret (2) as the zero vector of \(V^*\). Given \(s\in S\) and \(f\in V^*\) observe that \(f(\alpha _s)=1\) (resp. 0) if and only if the vertex \(s\) is assigned the on (resp. off ) state by \(f\).
For each \(s\in S\) define a linear transformation \(\kappa _s:V^*\rightarrow V^*\) by
$$\begin{aligned} \kappa _sf = f+f(\alpha _s)\sum _{st\in R}f_{t} \qquad \quad \mathrm{for\,\, all\,\, f} \in V^*. \end{aligned}$$
(3)
Fix a vertex \(s\) of \(\Gamma \). Given any \(f\in V^*\), if the state of \(s\) is on, then \(\kappa _sf\) is obtained from \(f\) by changing the states of all neighbors of \(s\), and \(\kappa _sf=f\) otherwise. Therefore, we may view \(\kappa _s\) as the move of the lit-only \(\sigma \)-game on \(\Gamma \) for which we choose the vertex \(s\) and change the states of all neighbors of \(s\) if the state of \(s\) is on. In particular \(\kappa _s^2=1\). For any vector space \(U\), let \(\mathrm{GL}(U)\) denote the general linear group of \(U\). Then \(\kappa _s\in \mathrm{GL}(V^*)\) for all \(s\in S\). The subgroup \(H=H_{\Gamma }\) of \(\mathrm{GL}(V^*)\) generated by the \(\kappa _s\) for all \(s\in S\) was first mentioned by Wu [19], which is called the flipping group of \(\Gamma \) in [12] and the lit-only group of \(\Gamma \) in [21].
The lit-only groups are closely related to the simply-laced Coxeter groups in the following way. Recall that the simply-laced Coxeter group \(W=W_{\Gamma }\) associated with \(\Gamma =(S,R)\) is the group generated by all elements \(s\in S\) subject to the relations
$$\begin{aligned} s^2&= 1,\\ (st)^2&= 1 \qquad \mathrm{if}\,\, st \not \in R,\\ (st)^3&= 1 \qquad \mathrm{if}\,\, st \in R \end{aligned}$$
for all \(s,t\in S\). By [12, Theorem 3.2], there exists a unique representation \(\kappa =\kappa _{\Gamma }:W\rightarrow \mathrm{GL}(V^*)\) such that \(\kappa (s)=\kappa _s\) for all \(s\in S\). Clearly \(\kappa (W)=H\). Given any \(f,g\in V^*\) observe that \(g\) can be obtained from \(f\) by a finite sequence of moves of the lit-only \(\sigma \)-game on \(\Gamma \) if and only if there exists \(w\in W\) such that \(g=\kappa (w)f\). Given an integer \(k\), the underlying graph \(\Gamma \) is \(k\)-lit if and only if for each \(\kappa (W)\)-orbit \(O\) on \(V^*\), there exists a subset \(K\) of \(S\) with size at most \(k\) such that \(\sum _{s\in K} f_s\in O\).
We now give the definitions of degenerate and nondegenerate graphs. Let \(B=B_{\Gamma }\) denote the symplectic form on \(V\) defined by
$$\begin{aligned} B(\alpha _s,\alpha _t)=\left\{ \begin{array}{ll} 1 &{}\quad \mathrm{if}\,\, st\in R,\\ 0 &{}\quad \mathrm{else} \end{array} \right. \end{aligned}$$
(4)
for all \(s,t\in S\) [16]. The radical of \(V\) (relative to \(B\)) is the subspace of \(V\) consisting of the vectors \(\alpha \) that satisfy \(B(\alpha ,\beta )=0\) for all \(\beta \in V\). The form \(B\) is said to be degenerate whenever the radical of \(V\) is nonzero and nondegenerate otherwise. The graph \(\Gamma \) is said to be degenerate whenever the form \(B\) is degenerate, and nondegenerate otherwise. The form \(B\) induces a linear map \(\theta :V\rightarrow V^*\) given by
$$\begin{aligned} \theta (\alpha )\beta =B(\alpha ,\beta ) \qquad \mathrm{for\,\, all}\,\, \alpha ,\beta \in V. \end{aligned}$$
(5)
Since the kernel of \(\theta \) is the radical of \(V\) and the matrix representing \(B\) with respect to the basis \(\{\alpha _s~|~s\in S\}\) is the adjacency matrix of \(\Gamma \) over \(\mathbb {F}_2\), the following lemma is straightforward.

Lemma 2.1

Let \(A\) denote the adjacency matrix of \(\Gamma \) over \(\mathbb {F}_2\). Then the following are equivalent:
  1. (i)

    \(\Gamma \) is a nondegenerate graph.

     
  2. (ii)

    \(\theta \) is an isomorphism of vector spaces.

     
  3. (iii)

    \(A\) is invertible.

     

Recall that given a simple graph \(G\), the line graph of\(G\) is a simple graph that has a vertex for each edge of \(G\), and two of these vertices are adjacent whenever the corresponding edges in \(G\) have a common vertex. The purpose of this paper is to investigate the lit-only \(\sigma \)-game on nondegenerate graphs which are not line graphs. Thus, it is natural to ask how to determine if a nondegenerate graph is a line graph. We will give two characterizations of nondegenerate line graphs as Proposition 2.4 below.

Lemma 2.2

Let \(G\) denote a finite simple connected graph of order \(n\). Assume that \(\Gamma \) is the line graph of \(G\). Then \(\theta (V)\) has dimension \(n-1\) if \(n\) is odd and has dimension \(n-2\) if \(n\) is even.

Proof

Let \(U\) denote the vertex space of \(G\) over \(\mathbb {F}_2\). Define a linear map \(\mu :V\rightarrow U\) by
$$\begin{aligned} \mu (\alpha _s)=u+v \qquad \mathrm{for\,\, all}\,\, s \in S, \end{aligned}$$
where \(u\) and \(v\) are the two endpoints of \(s\) in \(G\). Since \(G\) is connected, the image of \(\mu \) is the subspace of \(U\) consisting of these vectors each of which equals the sum of an even number of vertices of \(U\). Define a linear map \(\lambda :U\rightarrow V^*\) by
$$\begin{aligned} \lambda (u)\alpha _s = \left\{ \begin{array}{ll} 1 &{}\qquad \mathrm{if}\,\, u\,\, \mathrm{is\,\, incident\,\, to}\, s\, \mathrm{in}\, G,\\ 0 &{}\qquad \mathrm{else} \end{array} \right. \end{aligned}$$
for all \(u\in U\) and for all \(s\in S\). There is only one nonzero vector, the sum of all vertices of \(G\), in the kernel of \(\lambda \). Since \(\theta =\lambda \circ \mu \) and by the above comments, the result follows. \(\square \)

A claw is a tree with one internal vertex and three leaves. A simple graph is said to be claw-free if it does not contain a claw as an induced subgraph. A cut-vertex of \(\Gamma \) is a vertex of \(\Gamma \) whose deletion increases the number of components. A block of \(\Gamma \) is a maximal connected subgraph of \(\Gamma \) without cut-vertices. A block graph is a simple connected graph in which every block is a complete graph.

Lemma 2.3

[10, Theorem 8.5]. Let \(\Gamma \) denote a simple connected graph. Then \(\Gamma \) is the line graph of a tree if and only if \(\Gamma \) is a claw-free block graph.

The following proposition follows by combining Lemmas 2.1–2.3.

Proposition 2.4

Let \(\Gamma \) denote a simple connected graph. Then the following are equivalent:
  1. (i)

    \(\Gamma \) is a nondegenerate line graph.

     
  2. (ii)

    \(\Gamma \) is the line graph of an odd-order tree.

     
  3. (iii)

    \(\Gamma \) is a claw-free block graph of even order.

     

3 Main results

A quadratic form\(Q\)on\(V\) is a function \(Q:V\rightarrow \mathbb {F}_2\) satisfying
$$\begin{aligned} Q(\alpha +\beta )=Q(\alpha )+Q(\beta )+B(\alpha ,\beta ) \qquad \mathrm{for\,\, all}\,\, \alpha ,\beta \in V. \end{aligned}$$
(6)
Given a quadratic form \(Q\) on \(V\), the orthogonal group with respect to \(Q\) is the subgroup of \(\mathrm{GL}(V)\) consisting of all \(\sigma \in \mathrm{GL}(V)\) such that \(Q(\sigma \alpha )=Q(\alpha )\) for all \(\alpha \in V\). Given a basis \(P\) of \(V\) we define \(Q_P\) to be the unique quadratic form on \(V\) with \(Q_P(\alpha )=1\) for all \(\alpha \in P\).

For the rest of this paper, the form \(B\) is assumed to be nondegenerate. Moreover, let \(Q=Q_P\) where \(P=\{\alpha _s~|~s\in S\}\) and let \(O(V)\) denote the orthogonal group with respect to \(Q\). By (6), for any \(T\subseteq S\) a combinatorial interpretation of \(Q\big (\sum _{s\in T} \alpha _s\big )\) is the parity of the number of vertices and edges on the subgraph of \(\Gamma \) induced by \(T\).

We now can state the main results of this paper, which are Theorem 3.1, Theorem 3.2, and Corollary 3.3.

Theorem 3.1

Assume that \(\Gamma \) is a nondegenerate graph, but not a line graph. Then \(\kappa (W)\) is isomorphic to \(O(V)\). Moreover, the \(\kappa (W)\)-orbits on \(V^*\) are
$$\begin{aligned} \{0\}, \quad \theta (Q^{-1}(0){\setminus }\{0\}), \quad \theta (Q^{-1}(1)). \end{aligned}$$
Under the assumption that \(B\) is nondegenerate, the number \(|S|=2m\) is even and there exists a basis \(\{\beta _1,\gamma _1,\ldots ,\beta _m,\gamma _m\}\) of \(V\) such that \(B(\beta _i,\beta _j)=0,\, B(\gamma _i,\gamma _j)=0\) and
$$\begin{aligned} B(\beta _i,\gamma _j)= \left\{ \begin{array}{ll} 1 &{}\qquad \mathrm{if}\,\, i=j,\\ 0 &{}\qquad \mathrm{else} \end{array} \right. \end{aligned}$$
for all \(1\le i,j\le m\). Such a basis \(\{\beta _1,\gamma _1,\ldots ,\beta _m,\gamma _m\}\) of \(V\) is called a symplectic basis of \(V\). The Arf invariant of \(Q\) is defined to be
$$\begin{aligned} \mathrm{Arf}(Q)=\sum _{i=1}^m Q(\beta _i) Q(\gamma _i), \end{aligned}$$
which is independent of the choice of the symplectic basis \(\{\beta _1,\gamma _1,\ldots ,\beta _m,\gamma _m\}\) of \(V\) (for example see [1] or [9, Theorem 13.13]). Any two quadratic forms over \(\mathbb {F}_2\) are equivalent if and only if they have the same Arf invariant and the underlying spaces have the same dimension (for example see [1] or [9, Proposition 13.14]). The order of \(O(V)\) and the sizes of nontrivial \(O(V)\)-orbits on \(V\) are as follows (cf. [9, Chapter 14]). If \(\mathrm{Arf}(Q)=0\) then
$$\begin{aligned} \big |O(V)\big |&= 2^{m^2-m+1}(2^m-1)(2^2-1)(2^4-1)\cdots (2^{2m-2}-1),\\ \big |Q^{-1}(1)\big |&= 2^{2m-1}-2^{m-1},\\ \big |Q^{-1}(0){\setminus }\{0\}\big |&= 2^{2m-1}+2^{m-1}-1. \end{aligned}$$
If \(\mathrm{Arf}(Q)=1\) then
$$\begin{aligned} \big |O(V)\big |&= 2^{m^2-m+1}(2^{m}+1)(2^2-1)(2^4-1)\cdots (2^{2m-2}-1),\\ \big |Q^{-1}(1)\big |&= 2^{2m-1}+2^{m-1},\\ \big |Q^{-1}(0){\setminus }\{0\}\big |&= 2^{2m-1}-2^{m-1}-1. \end{aligned}$$
For each \(s\in S\), there exists \(\alpha _s^\vee \in V\) such that
$$\begin{aligned} B(\alpha _s^\vee ,\alpha _t)=\left\{ \begin{array}{ll} 1 &{}\qquad \mathrm{if}\,\, s=t,\\ 0 &{}\qquad \mathrm{else} \end{array} \right. \end{aligned}$$
(7)
for all \(t\in S\). The set \(\{\alpha _s^\vee |s\in S\}\) forms a basis of \(V\) and is called the basis of\(V\)dual to\(\{\alpha _s~|~s\in S\}\) (with respect to\(B\)).

Theorem 3.2

Assume that \(\Gamma =(S,R)\) is a nondegenerate graph, but not a line graph. Then \(\Gamma \) is 2-lit. Moreover, the following are equivalent:
  1. (i)

    \(\Gamma \) is 1-lit.

     
  2. (ii)

    The restriction of \(Q\) to \(\{\alpha _s^\vee ~|~s\in S\}\) is surjective.

     

When the nondegenerate graph \(\Gamma \) is bipartite, Theorem 3.2 can be improved as follows.

Corollary 3.3

Assume that \(\Gamma \) is a nondegenerate bipartite graph. Then \(\Gamma \) is 2-lit. Moreover, the following are equivalent:
  1. (i)

    \(\Gamma \) is 1-lit

     
  2. (ii)

    \(\Gamma \) contains a vertex with even degree or \(\Gamma \) is a single edge.

     
As consequences of Corollary 3.3, we obtain two families of 1-lit graphs as follows.
  • A tree is nondegenerate if and only if it has a perfect matching. By [11, Lemma 2.4], a tree with a perfect matching satisfies Corollary 3.3(ii) and is therefore 1-lit (cf. [14, Theorem 1.1]). This result gives a partial affirmative answer for [20, Conjecture 7].

  • For any two positive integers \(m\) and \(n\), the \(m\times n\) grid is nondegenerate if and only if \(m+1\) and \(n+1\) are coprime [18]. By Corollary 3.3 any such \(m\times n\) grid is 1-lit. This result partially improves [8, Theorem 26].

The following example shows that Corollary 3.3 is no longer true if the assumption of \(\Gamma \) is the same as that of Theorem 3.2. Consider the graph \(\Gamma =(S,R)\) as below.
The graph \(\Gamma =(S,R)\) is nondegenerate and not a block graph. Therefore \(\Gamma \) is not a line graph by Proposition 2.4. The basis \(\{\alpha _1^\vee ,\alpha _2^\vee ,\ldots ,\alpha _6^\vee \}\) of \(V\) dual to \(\{\alpha _1,\alpha _2,\ldots ,\alpha _6\}\) can be expressed as follows.
$$\begin{aligned} \begin{array}{ll} \alpha _1^\vee =\alpha _2+\alpha _6, \qquad \quad &{}\alpha _4^\vee =\alpha _3+\alpha _5,\\ \alpha _2^\vee =\alpha _1+\alpha _3+\alpha _5+\alpha _6, \qquad \quad &{}\alpha _5^\vee =\alpha _2+\alpha _3+\alpha _4+\alpha _6, \qquad \quad \\ \alpha _3^\vee =\alpha _2+\alpha _4+\alpha _5, \qquad \quad &{}\alpha _6^\vee =\alpha _1+\alpha _2+\alpha _5. \end{array} \end{aligned}$$
A direct computation shows that \(Q(\alpha _s^\vee )=0\) for all \(s\in S\). Therefore \(\Gamma \) is not 1-lit by Theorem 3.2, but the vertices 2,5 have even degree in \(\Gamma \).

4 Proof of Theorem 3.1

To prove Theorem 3.1, we consider a family of linear transformations on \(V\) defined as follows. For \(\alpha \in V\), the transvection on\(V\)with direction\(\alpha \) is a linear transformation \(\tau _\alpha :V\rightarrow V\) defined by
$$\begin{aligned} \tau _\alpha \beta = \beta +B(\beta ,\alpha )\alpha \qquad \mathrm{for\,\, all}\,\, \beta \in V. \end{aligned}$$
Observe that \(\tau _\alpha \) preserves the form \(B\) and that \(\tau _\alpha \in \mathrm{GL}(V)\) since \(\tau _\alpha ^2=1\).

For a subset \(P\) of \(V\) define \(Tv(P)\) to be the subgroup of \(\mathrm{GL}(V)\) generated by \(\tau _\alpha \) for \(\alpha \in P\), and define \(G(P)\) to be the simple graph whose vertex set is \(P\) and where \(\alpha ,\beta \) in \(P\) form an edge if and only if \(B(\alpha ,\beta )=1\). For any two linearly independent sets \(P\) and \(P'\) of \(V\), we say that \(P'\) is elementary t-equivalent to \(P\) whenever there exist \(\alpha ,\beta \in P\) such that \(P'\) is obtained from \(P\) by changing \(\beta \) to \(\tau _\alpha \beta \). The equivalence relation generated by the elementary \(t\)-equivalence relation is called the t-equivalence relation [3].

Lemma 4.1

[3, Theorem 3.3]. Let \(P\) denote a linearly independent set of \(V\). Assume that \(G(P)\) is a connected graph. Then there exists \(P'\) in \(t\)-equivalence class of \(P\) for which \(G(P')\) is a tree.

Lemma 4.2

[15, Lemma 3.7]. Let \(P\) denote a linearly independent set of \(V\). Assume that \(G(P)\) is the line graph of a tree. Then, for each \(P'\) in the \(t\)-equivalence class of \(P\), the graph \(G(P')\) is the line graph of a tree.

A basis \(P\) of \(V\) is said to have orthogonal type [4] if \(P\) is \(t\)-equivalent to some \(P'\) for which \(G(P')\) is a tree containing the graph

as a subgraph.

Lemma 4.3

Assume that \(P\) is a basis of \(V\) for which \(G(P)\) is a tree, but not a path. Then \(P\) is of orthogonal type.

Proof

Since \(G(P)\) is not a path it contains a vertex \(\alpha \) with degree at least three. If any two neighbors of \(\alpha \), say \(\beta \) and \(\gamma \), are leaves of \(G(P)\), then \(\beta +\gamma \) lies in the radical of \(V\), which contradicts that \(B\) is nondegenerate. Therefore, at most one neighbor of \(\alpha \) is a leaf in \(G(P)\) and so \(P\) is of orthogonal type.

Lemma 4.4

[4, Section 10]. Let \(P\) denote a basis of \(V\) which is of orthogonal type. Then \(Tv(P)\) is the orthogonal group with respect to \(Q_P\). Moreover, the \(Tv(P)\)-orbits on \(V\) are
$$\begin{aligned} \{0\}, \quad Q_P^{-1}(0){\setminus }\{0\}, \quad Q_P^{-1}(1). \end{aligned}$$

Proof of Theorem 3.1

For each \(s\in S\), let \(\tau _s\) denote the transvection on \(V\) with direction \(\alpha _s\). By [16, Section 5], there exists a unique representation \(\tau =\tau _{\Gamma }:W\rightarrow \mathrm{GL}(V)\) such that \(\tau (s)=\tau _s\) for all \(s\in S\). For each \(w\in W\) the transpose of \(\tau (w^{-1})\) is equal to \(\kappa (w)\). Therefore \(\kappa \) is the dual representation of \(\tau \). Since \(\tau \) preserves the form \(B\) we have
$$\begin{aligned} \theta \circ \tau (w)=\kappa (w)\circ \theta \qquad \quad \mathrm{for\,\,all}\,\, w \in W. \end{aligned}$$
(8)
Let \(P=\{\alpha _s~|~s\in S\}\). Clearly \(Tv(P)=\tau (W)\) and \(G(P)\) is (isomorphic to) \(\Gamma \). By Lemma 4.1 there exists \(P'\) in \(t\)-equivalence class of \(P\) for which \(G(P')\) is a tree. Since \(G(P)\) is not a line graph, the tree \(G(P')\) is not a path by Lemma 4.2. By Lemma 4.3 the basis \(P'\) of \(V\), as well as \(P\), is of orthogonal type. By Lemma 4.4, the group \(\tau (W)=O(V)\) and the \(\tau (W)\)-orbits on \(V\) are \(\{0\}\), \(Q^{-1}(0){\setminus }\{0\}\), and \(Q^{-1}(1)\). Applying (8) and since \(\theta \) is an isomorphism by Lemma 2.1, the result follows. \(\square \)

5 Proof of Theorem 3.2 and Corollary 3.3

Recall the basis \(\{\alpha ^\vee _s~|~s\in S\}\) of \(V\) from (7). To prove Theorem 3.2 and Corollary 3.3, we introduce a simple graph which includes the information of the values \(B(\alpha ^\vee _s,\alpha ^\vee _t)\) for all \(s,t\in S\) as follows.

Define \(R^\vee \) to be the set consisting of all two-element subsets \(\{s,t\}\) of \(S\) with \(B(\alpha _s^\vee ,\alpha _t^\vee )=1\). Define \(\Gamma ^\vee \) to be the simple graph with vertex set \(S\) and edge set \(R^\vee \). We will refer to \(\Gamma ^\vee \) as the dual graph of \(\Gamma \). Note that the notion of dual graphs defined above is different from the usual ones in graph theory. The following lemma suggests why the graph \(\Gamma ^\vee \) is of interest.

Lemma 5.1

For each \(s\in S\) we have \(\theta (\alpha _s^\vee )=f_s\).

Proof

Let \(s,t\in S\) be given. Using (5) and (7), we have \(\theta (\alpha _s^\vee )\alpha _t=1\) whenever \(s=t\) and otherwise \(\theta (\alpha _s^\vee )\alpha _t=0\). Comparing this with (1) the result follows. \(\square \)

Lemma 5.2

For each \(s\in S\) we have
$$\begin{aligned} \alpha _s=\sum _{st\in R}\alpha _t^\vee . \end{aligned}$$

Proof

Fix \(s\in S\). By (1), (4), and (5), the vector \(\theta (\alpha _s)\) is equal to
$$\begin{aligned} \sum _{st\in R}f_t. \end{aligned}$$
By Lemma 5.1 the above is equal to
$$\begin{aligned} \theta \bigg (\sum _{st\in R}\alpha _t^\vee \bigg ). \end{aligned}$$
Now, by Lemma 2.1(ii) this lemma follows. \(\square \)

Observe that \(B_{\Gamma ^\vee }\) is equivalent to \(B\). Therefore \(\Gamma ^\vee \) is a nondegenerate graph. Since \(\{\alpha _s~|~s\in S\}\) is the basis of \(V\) dual to \(\{\alpha _s^\vee ~|~s\in S\}\), the graph \(\Gamma \) is the dual graph of \(\Gamma ^\vee \). By duality Lemma 5.2 implies that

Lemma 5.3

For each \(s\in S\) we have
$$\begin{aligned} \alpha _s^\vee =\sum _{st\in R^\vee }\alpha _t. \end{aligned}$$

Lemma 5.4

Let \(A\) and \(A^\vee \) denote the adjacency matrices of \(\Gamma \) and \(\Gamma ^\vee \) over \(\mathbb {F}_2\), respectively. Then \(A\) and \(A^\vee \) are inverses of each other.

Proof

We show that \(A^\vee A\) is equal to the identity matrix. Let \(s,t\in S\) be given. By the comment below Lemma 5.1 the \((s,t)\)-entry of \(A\) (resp. \(A^\vee \)) is equal to \(B(\alpha _s,\alpha _t)\) (resp. \(B(\alpha _s^\vee ,\alpha _t^\vee )\)). By the definition of \(\Gamma ^\vee \) the \((s,t)\)-entry of \(A^\vee A\) is equal to
$$\begin{aligned} B\bigg (\sum _{su\in R^\vee }\alpha _u,\alpha _t\bigg ). \end{aligned}$$
(9)
By Lemma 5.3 the vector in the first coordinate of (9) is equal to \(\alpha _s^\vee \). Therefore (9) is equal to 1 if and only if \(s=t\) by (7). The result follows. \(\square \)

We are now ready to prove Theorem 3.2.

Proof of Theorem 3.2

In Lemma 5.1 we saw that \(\theta (\alpha _s^\vee )=f_s\) for all \(s\in S\). Therefore (i) and (ii) are equivalent by Theorem 3.1. To show that \(\Gamma \) is 2-lit, it is now enough to consider the two cases: (a) \(Q(\alpha _s^\vee )=0\) for all \(s\in S;\) (b) \(Q(\alpha _s^\vee )=1\) for all \(s\in S\).
  1. (a)

    It suffices to show that there exist \(s,t\in S\) such that \(Q(\alpha _s^\vee +\alpha _t^\vee )=1\). Since the form \(B\) is nontrivial there exist \(s,t\in S\) such that \(B(\alpha _s^\vee ,\alpha _t^\vee )=1\). Then the \(s\) and \(t\) are the desired elements in \(S\).

     
  2. (b)

    It suffices to show that there exist two distinct \(s,t\in S\) such that \(Q(\alpha _s^\vee +\alpha _t^\vee )=0\). By our assumption, the graph \(\Gamma \) is not a complete graph. Using Lemma 5.4, we deduce that \(\Gamma ^\vee \) is not a complete graph. Therefore there exist two distinct \(s,t\in S\) such that \(B(\alpha _s^\vee ,\alpha _t^\vee )=0\). Such \(s\) and \(t\) are the desired elements in \(S\).\(\square \)

     

To prove Corollary 3.3, we give a sufficient condition for Theorem 3.2(ii).

Lemma 5.5

Let \(\Gamma =(S,R)\) denote a nondegenerate graph. Assume that there exists \(s\in S\) with even degree in \(\Gamma \) such that
$$\begin{aligned} \mathop {\sum _{\{u,v\}\subseteq S}}\limits _{su,sv\in R} B(\alpha _u^\vee ,\alpha _v^\vee )=0, \end{aligned}$$
(10)
where the sum is over all two-element subsets \(\{u,v\}\) of \(S\) with \(su,sv\in R\). Then the restriction of \(Q\) to \(\{\alpha _t^\vee ~|~st\in R\}\) is surjective.

Proof

Apply \(Q\) to either side of the equation in Lemma 5.2. Using (6), (10) and \(Q(\alpha _s)=1\) to evaluate the resulting equation, we obtain that
$$\begin{aligned} \sum _{st\in R}Q(\alpha _t^\vee )=1. \end{aligned}$$
(11)
By (11) there exists a neighbor \(u\) of \(s\) for which \(Q(\alpha _u^\vee )=1\). Since \(s\) has even degree in \(\Gamma \) there exists a neighbor \(v\) of \(s\) for which \(Q(\alpha _v^\vee )=0\). The result follows. \(\square \)

Proof of Corollary 3.3

By Proposition 2.4 a nondegenerate bipartite graph \(\Gamma \) is a line graph if and only if \(\Gamma \) is a path of even order. Since every path is 1-lit, this corollary holds for \(\Gamma \) as a line graph. We thus assume that \(\Gamma \) is not a line graph. By Theorem 3.2 the graph \(\Gamma \) is 2-lit. By Lemma 5.4 we deduce that the graph \(\Gamma ^\vee \) is bipartite with the same bipartition that of \(\Gamma \). We use this to show that (i) and (ii) are equivalent.
(ii) \(\Rightarrow \) (i):

Let \(s\) denote a vertex of \(\Gamma \) with even degree. Since \(\Gamma \) and \(\Gamma ^\vee \) are bipartite graphs with same bipartition, we deduce that \(B(\alpha _u^\vee ,\alpha _v^\vee )=0\) for any neighbors \(u,v\) of \(s\) in \(\Gamma \). Therefore (10) holds. By Lemma 5.5 the restriction of \(Q\) on \(\{\alpha _t^\vee ~|~st\in R\}\) is onto. Therefore \(\Gamma \) is 1-lit by Theorem 3.2.

(i) \(\Rightarrow \) (ii):
Suppose on the contrary that each vertex of \(\Gamma \) has odd degree. Using Lemma 5.4, we deduce that each vertex of \(\Gamma ^\vee \) has odd degree. Let \(s\) denote any element of \(S\). By Lemma 5.3, \(Q(\alpha _s^\vee )\) is equal to
$$\begin{aligned} Q\bigg (\sum _{st\in R^\vee }\alpha _t \bigg ). \end{aligned}$$
(12)
Since the bipartite graphs \(\Gamma \) and \(\Gamma ^\vee \) have the same bipartition, we deduce that \(B(\alpha _u,\alpha _v)=0\) for any neighbors \(u,v\) of \(s\) in \(\Gamma ^\vee \). By (6), the summation in (12) can be moved out front. Since \(Q(\alpha _s)=1\) for all \(s\in S\), it follows that (12) is equal to 1, contradicting Theorem 3.2(ii). \(\square \)

Notes

Acknowledgments

This work was supported by National Center of Theoretical Sciences and Ministry of Science and Technology of Taiwan.

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Authors and Affiliations

  1. 1.Mathematics DivisionNational Center for Theoretical Sciences, National Tsing-Hua UniversityHsinchuTaiwan, ROC

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