# Lit-only sigma-game on nondegenerate graphs

## 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

*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

*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 \(f\).

*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].

*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

**Lemma 2.1**

- (i)
\(\Gamma \) is a nondegenerate graph.

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

- (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*

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**

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

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

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

## 3 Main results

*quadratic form*\(Q\)

*on*\(V\) is a function \(Q:V\rightarrow \mathbb {F}_2\) satisfying

*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**

*symplectic basis*of \(V\). The

*Arf invariant*of \(Q\) is defined to be

*the basis of*\(V\)

*dual to*\(\{\alpha _s~|~s\in S\}\) (

*with respect to*\(B\)).

**Theorem 3.2**

- (i)
\(\Gamma \) is 1-lit.

- (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**

- (i)
\(\Gamma \) is 1-lit

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

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].

## 4 Proof of Theorem 3.1

*transvection on*\(V\)

*with direction*\(\alpha \) is a linear transformation \(\tau _\alpha :V\rightarrow V\) defined by

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**

*Proof of Theorem 3.1*

## 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**

*Proof*

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**

**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 are now ready to prove Theorem 3.2.

*Proof of Theorem 3.2*

- (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 \(S\).

- (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**

*Proof*

*Proof of Corollary 3.3*

- (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)

## Notes

### Acknowledgments

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

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