Litonly sigmagame on nondegenerate graphs
Abstract
A configuration of the litonly \(\sigma \)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 litonly \(\sigma \)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 litonly \(\sigma \)game on nondegenerate graphs \(\Gamma \) which are not line graphs. We show that these graphs \(\Gamma \) are 2lit and provide a linear algebraic criterion for \(\Gamma \) to be 1lit.
Keywords
Group action Litonly \(\sigma \)game Nondegenerate graphMathematics Subject Classification
Primary 05C57 Secondary 15A63 20F551 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: litonly \(\sigma \) game. Starting from an initial configuration, the goal of the litonly \(\sigma \)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 litonly \(\sigma \)game on \(\Gamma \) and the smallest integer \(k\), the minimum light number of \(\Gamma \) [19], for which \(\Gamma \) is \(k\)lit. The notion of litonly \(\sigma \)games occurred implicitly in the study of equivalence classes of Vogan diagrams. The Borelde Siebenthal theorem [2] showed that every Vogan diagram is equivalent to one with a singlepainted vertex, which implies that each simplylaced 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 “litonly \(\sigma \)game” was coined.
The litonly \(\sigma \)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 litonly \(\sigma \)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 litonly \(\sigma \)game on a simple graph \(\Gamma \) can also be considered as a representation \(\kappa _{\Gamma }\) of the simplylaced 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 litonly \(\sigma \)game on \(\Gamma \) (Theorem 3.1). Moreover, we show that these graphs \(\Gamma \) are 2lit and provide a linear algebraic criterion for \(\Gamma \) to be 1lit (Theorem 3.2). Combining Theorem 3.1, Theorem 3.2, and those in [13, 21], the study of the litonly \(\sigma \)game on nondegenerate graphs is quite completed, and the focus for further research is on degenerate graphs.
2 Preliminaries
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 litonly \(\sigma \)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 \(n1\) if \(n\) is odd and has dimension \(n2\) if \(n\) is even.
Proof
A claw is a tree with one internal vertex and three leaves. A simple graph is said to be clawfree if it does not contain a claw as an induced subgraph. A cutvertex 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 cutvertices. 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 clawfree 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 oddorder tree.
 (iii)
\(\Gamma \) is a clawfree block graph of even order.
3 Main results
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
Theorem 3.2
 (i)
\(\Gamma \) is 1lit.
 (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 1lit
 (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 1lit (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 1lit. This result partially improves [8, Theorem 26].
4 Proof of Theorem 3.1
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 tequivalent 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 tequivalence 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 twoelement 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 1lit 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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