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.

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:

• (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

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:

• (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

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:

• (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

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

• (i) $\Gamma$ is 1-lit

• (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 $s\in S$.

• (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

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). $\square$