The Gonality of Chess Graphs

The game of chess has a long history of inspiring problems in math. How many queens (or bishops, or knights) can be placed on a chessboard (not necessarily $8\times 8$) such that no pair of them can attack each other? Similarly, what is the smallest number of queens (or bishops, or knights) that can be placed so that every square on the board may be attacked in one move? Can a knight visit each square on an $m\times n$ chessboard exactly once, returning to its starting position? These questions can be framed and studied in the language of graph theory, and relate to such topics as independence number, domination number, and Hamiltonian cycles.

Choose a chess piece, either king, queen, rook, bishop, or knight¹¹1We omit pawns from consideration, since the movement of a pawn is not symmetric between two squares, so an undirected graph does not suffice to encode their movements,. The $m\times n$ chess graph associated to that piece has $mn$ vertices, corresponding to the squares of an $m\times n$ chessboard. Two vertices are connected by an edge if and only if the chess piece in question can move between the two corresponding squares. For example, in a king’s graph, each vertex is adjacent to at most eight other vertices, corresponding to two horizontal, two vertical, and four diagonal moves. Toroidal chess graphs are constructed similarly, with moves allowed that glue the boundaries of the chessboard to form a torus. Illustrated on the left in Figure 1 is a $4\times 5$ chessboard with a king on one of the squares. The king’s moves from that square under traditional chess rules are illustrated with solid arrows; when toroidal moves are allowed, three more moves are possible, illustrated with dashed arrows. On the right are the $20$ vertices of the $4\times 5$ king’s graph; the edges coming from one vertex are illustrated, with dashed edges for those only present in the toroidal king’s graph.

Note that on a toroidal board, a piece may be able to travel between two vertices with multiple routes; for example, on a $2\times n$ chess board, a knight moving two squares spots up and one the right has the same effect as moving two squares down and one to the right. We follow the convention that only one edge connects the two corresponding vertices, although an interesting variant of our problems would be to encode such repeated moves with repeated edges (turning our graph into a multigraph). For some graphs, like rook’s graphs, making the board toroidal has no impact at all; for others, such as bishop’s graphs, making the board toroidal increases the number of edges and changes the graph structure drastically.

In this paper we study chess graphs through the lens of chip-firing games and graph gonality. The (divisorial) gonality of a graph, a discrete analog of the gonality of an algebraic curve, is the minimum degree of a positive rank divisor on that graph. Phrased in the language of chip-firing games, the gonality of a graph $G$ is the smallest number of chips that can be placed on $G$ so that a chip can be placed on any vertex through chip-firing moves, without any other vertex being in debt. This invariant has been of great interest due to its connection to algebraic geometry [3], and much work has been done to determine the gonality of different families of graphs.

Several families of chess graphs have already been the subject of study from this perspective. The gonality of rook’s graphs was first explored in [2], with gonality determined for $m\times n$ graphs with $\min\{m,n\}\leq 5$. This result was generalized in [14], where gonality was determined for all rook’s graphs. More recently, the gonality of queen’s graphs and toroidal queen’s graphs was determined in [12]. In this paperwe study the remaining chess graphs, namely the traditional and toroidal versions of king’s, bishop’s, and knight’s graphs arising from $m\times n$ chessboards. The $3\times 4$ non-toroidal versions of these graphs are illustrated in the first row Figure 2; the second row shows the edges that are gained by the upper left vertex when expanding to the toroidal version.

We compute gonality exactly for all toroidal bishop’s graphs, as well as for king’s and toroidal king’s graphs where the number of columns is sufficiently high compared to the number of rows. We also find upper and lower bounds on gonality for king’s, toroidal king’s, bishop’s, knight’s, and toroidal knight’s graphs.

We summarize our results in the following two theorems; in both we assume $m\leq n$. The first enumerates our contribution to known bounds on gonality.

The second theorem gives a qualitative summary of how different graphs behave as we fix $m$ and vary $n$.

Our paper is organized as follows. In Section 2 we present background and definitions. In Section 3 we present our results on king’s graphs, followed by toroidal king’s graphs in Section 4. In Section 5 we present our results on bishop’s graphs, followed by toroidal bishop’s graphs in Section 6. In Section 7 we present our results on knight’s graphs, followed by toroidal knight’s graphs in Section 8. We present directions for future work in Section 9, and several computations referred to in earlier sections in the Appendices.

Let $G$ be a graph consisting of a finite set $V=V(G)$ of vertices and a finite multiset $E=E(G)\subseteq\{\{v,w\}\,|\,v,w\in V,v\neq w\}$ of edges. If $E$ has no repeated edges, we call $G$ simple graph. If a vertex $v$ is in an edge $e$, we say $e$ is incident to $v$, and if $\{v,w\}\in E$, we say $v$ and $w$ are adjacent. We refer to the number of edges incident to $v$ as the valence²²2This is also referred to as the degree of $v$; however, the term degree is used with a different meaning in chip-firing games. of $v$, denoted $\operatorname{val}(v)$. For disjoint subsets $A,B\subset E(G)$, we let $E(A,B)$ be the set of edges with a vertex in $A$ and a vertex in $B$; and we let $E(v,w)=E(\{v\},\{w\})$. If for all pairs of vertices $v,w\in V$ we can find vertices $v=v_{0},v_{1},v_{2},\ldots,v_{k}=w$ such that $v_{i}$ and $v_{i+1}$ are adjacent for all $i$, we say $G$ is connected; otherwise it is disconnected. A subset $V^{\prime}\subset V(G)$ of vertices is said to be connected if the subgraph $G^{\prime}$ with vertex set $V^{\prime}$ and edge set $\{\{v,w\}\in E\,|\,v,w\in V^{\prime}\}$ is connected graph.

An $m\times n$ chess graph has vertex set

When it is more notationally convenient, we may denote the vertex $(i,j)$ as $v_{i,j}$. We depict these vertices in a grid, with $m$ rows and $n$ columns, treating the upper left corner as the vertex $(1,1)$, with $i$ denoting the row of the vertex and $j$ denoting the column. Identifying these vertices with the squares in an $m\times n$ chess board, two vertices are adjacent if and only if the chess piece in question can move directly between the corresponding squares. In particular, we have the following adjacency rules for when distinct vertices $(i_{1},j_{1})$ and $(i_{2},j_{2})$ are adjacent in our chess graphs.

• •

  King’s graph: when $i_{1}-i_{2},j_{1}-j_{2}\in\{-1,0,1\}.$

• •

  Bishop’s graph: when $(i_{1}-i_{2},j_{1}-j_{2})$ is an integer multiple of either $(1,1)$ or $(1,-1)$.

• •

  Knight’s graph: when $(i_{1}-i_{2},j_{1}-j_{2})\in\{(\pm 1,\pm 2),(\pm 2,\pm 1)\}$.

• •

  Rook’s graph: when $i_{1}=i_{2}$ or $j_{1}=j_{2}$.

• •

  Queen’s graph: when either $(i_{1}-i_{2},j_{1}-j_{2})$ is an integer multiple of either $(1,1)$ or $(1,-1)$, or $i_{1}=i_{2}$ or $j_{1}=j_{2}$.

An $m\times n$ toroidal chess graph has the same vertex set and the same adjacency rules, except that all equalities are considered modulo $m$ for the first coordinate and modulo $n$ for the second coordinate. When representing these graphs in figures, we either draw the graph with vertices and edges, or when such a figure would be hard to digest visually simply present a chessboard, with a caption making clear which chess piece we are considering.

We now present the theory of divisors on graphs, and refer the reader to [7] for more details. We remark that most treatments of chip-firing games assume our graph is connected; we do not assume that here, and will note when our difference in assumptions has an impact on various results.

Given a graph $G$, a divisor is a map $D:V(G)\rightarrow\mathbb{Z}$, i.e. an assignment of an integer to each vertex of $G$. A divisor is thought of intuitively as a collection of poker chips placed on each vertex, where $v$ receives $D(v)$ chips, with negative integers representing debt. If $D(v)\geq 0$ for all $v$, we say that $D$ is effective. The total number of chips on the graph, $\sum_{v\in V(G)}D(v)$, is referred to as the degree of the divisor, denoted $\deg(D)$. The set of all divisors forms a group under the operation defined by vertex-wise addition (i.e. $(D+D^{\prime})(v)=D(v)+D^{\prime}(v)$; this group is isomorphic to the free abelian group on $V(G)$.

Given a divisor $D$ on a graph $G$ and a vertex $v\in V(G)$, we can transform $D$ into another divisor $D^{\prime}$ through a chip-firing move at $v$. This removes $\operatorname{val}(v)$ chips from $v$ and transfers them to those vertices adjacent to $v$, one along each edge incident to $v$. More formally, $D^{\prime}(v)=v-\operatorname{val}(v)$, and $D^{\prime}(w)=D(w)+|E(v,w)|$ for all $w\neq v$. Two divisors $D$ and $D^{\prime}$ are said to be equivalent, written $D\sim D^{\prime}$, if $D^{\prime}$ can be obtained from $D$ via some sequence of chip-firing moves. This relation is an equivalence relation on the set of all divisors.

A collection of vertices can be fired in any order without changing the final outcome. This leads us to define set-firing moves: given $A\subset V(G)$, set-firing $A$ means to fire the vertices of $A$ in any order. This results in a net flow of chips only along the edges leaving the set, whereas for $v,w\in A$ the movement of chips from $v$ to $w$ cancels with the movement of chips from $w$ to $v$. Examples of two set-firing moves are depicted in Figure 3. Since the divisors differ by a sequence of chip-firing moves, all three are equivalent.

We now define the rank of a divisor $D$, denoted $r(D)$. If $D$ is not equivalent to any effective divisor, we say $r(D)=-1$. Otherwise, $r(D)$ is the maximum integer $k$ such that, for every effective divisor $D^{\prime}$ with $\deg(D^{\prime})=k$, the divisor $D-D^{\prime}$ is equivalent to some effective divisor. Intuitively, we can think of the rank of a divisor as the maximum amount of added debt that divisor can eliminate from the graph, regardless of how that debt is placed.

The (divisorial) gonality of a graph $G$, denoted $\operatorname{gon}(G)$, is the minimum degree of a positive rank divisor on a graph, i.e. the minimum degree of a divisor that can eliminate $-1$ debt from any vertex without introducing debt elsewhere. We can equivalently define $\operatorname{gon}(G)$ as the minimum degree of a divisor $D$ such that for any vertex $v\in V(G)$, there exists $D^{\prime}\sim D$ with $D^{\prime}$ effective and $D^{\prime}(v)\geq 1$.

For example, the divisor illustrated in 3 has positive rank: firing the first column, then the first two, then the first three, and so on can move a chip to any vertex without introducing debt elsewhere. Since that divisor has degree $4$, it follows that the gonality of that graph is at most $4$.

While an explicit positive rank divisor provides an upper bound on gonality, finding a lower bound is more difficult. A priori, to show that the gonality of a graph is at least $k$, one must consider all degree $k-1$ divisors, and show that for each there exists at least one vertex $v$ such that that divisor cannot eliminate debt on $v$. While extremely brute force, this strategy can be implemented for relatively small graphs by leveraging Dhar’s burning algorithm [7], which refers to the first three steps of the following process.

1. (1)

   Given a graph $G$ and an effective divisor $D$, choose a vertex $q\in G$. Start a “fire” at this vertex, burning it.

2. (2)

   Whenever an edge is incident to a burned vertex, that edge burns.

3. (3)

   Whenever a vertex $v$ is incident to more than $D(v)$ burning edges, the vertex $v$ burns.

4. (4)

   If the whole graph burns, terminate. If some set $S$ of vertices is unburned, set-fire $S$, and return to step (1).

An example of the start of this algorithm is illustrated Figure 4. The fire starts at the vertex $q$, and spreads along incident edges. The adjacent vertex with $1$ chip does not immediately burn, though the one with $0$ chips does. Its edges burn, burning the left vertex with one chip, burning its edges. Here the fire stabilizes, since the two rightmost vertices have at least as many chips as burning edges. Those two vertices are fired, yielding the divisor on the right. If we ran the burning process again, the whole graph would burn (first the left vertex with $1$ chip, then the vertex with $2$ chips, and from there the rest of the graph). Since the whole graph would burn with no chips on $q$, we know that the illustrated divisor does not have positive rank.

One can show in general that firing $S$ never introduces debt, and that this process will eventually terminate; see [7, Chapter 3] for details. Moreover, if we start with $q$ such that $D(q)=0$, then the final divisor places a chip on $q$ if and only if there exists an effective divisor $D^{\prime}\sim D$ such that $D^{\prime}(q)\geq 1$. Thus this method can be used to check if an effective divisor $D$ has positive rank: for each $v\in V(G)$, run the algorithm from $v$, and check if $v$ ends up with at least one chip. If yes for all $v$, then $r(D)>0$; otherwise $r(D)=0$. From there, showing that $\operatorname{gon}(G)\geq k$ is accomplished by performing this check for all effective divisors of degree $k-1$.

To avoid such a brute-force approach, one can study graph parameters that serve as lower bounds on gonality. For instance, the well-studied parameter of treewidth, $\operatorname{tw}(G)$, is known to be a lower bound $\operatorname{gon}(G)$ [16]. More recently, the scramble number of a graph, $\mathrm{sn}(G)$, was introduced in [11] and shown to be a lower bound on $\operatorname{gon}(G)$. Since $\operatorname{tw}(G)\leq\mathrm{sn}(G)\leq\operatorname{gon}(G)$ for any graph $G$ [11, Theorem 1.1], scramble number is a strictly better lower bound, so we focus on it here.

A scramble $\mathcal{S}$ on a graph $G$ is a nonempty collection of connected subsets (referred to as eggs) of $V(G)$. To any scramble $\mathcal{S}$, we associate two numbers: the hitting number $h(\mathcal{S})$, and the egg-cut number $e(\mathcal{S})$. A hitting set for $\mathcal{S}$ is a set $V^{\prime}\subset V(G)$ such that every egg in $\mathcal{S}$ has a vertex in $V^{\prime}$; $h(\mathcal{S})$ is then the minimum size of any hitting set. An egg-cut for $\mathcal{S}$ is a subset $E^{\prime}\subset E(G)$ such that deleting $E^{\prime}$ from $G$ disconnects the graph into multiple components, at least two of which contain an egg; $e(\mathcal{S})$ is then the minimum size of any egg-cut. The order of a scramble is defined to be the minimum of the hitting number and the egg-cut number:

Finally, the scramble number of a graph is the maximum order of any scramble on the graph.

For an example of a scramble, consider the $4\times 5$ grid graph in Figure 3. Let $\mathcal{S}$ consist of five eggs, the first consisting of the four vertices in the first column, the second consisting of the four vertices in the second column, and so on. Since these eggs do not overlap, we immediately have the the hitting number equals the number of eggs, so $h(\mathcal{S})=5$. The four edges connecting the first and second columns is an egg-cut, so $e(\mathcal{S})\leq 4$. To see that $e(\mathcal{S})\geq 4$, consider an arbitrary pair of eggs in $\mathcal{S}$. We can find four pairwise edge-disjoint paths between them: start at any vertex of the left egg and moving across the horizontal edges until reaching the right egg. Any egg-cut $E^{\prime}$ separating those two eggs must include at least one edge from each of these four paths; otherwise the eggs would be in the same component of $G-E^{\prime}$. As our choice of eggs was arbitrary, we have that any egg-cut has at least four edges in it, so $e(\mathcal{S})\geq 4$, and thus $e(\mathcal{S})=4$. This means that $||\mathcal{S}||=\min\{4,5\}=4$, so $\mathrm{sn}(G)\geq 4$. In fact, since $\operatorname{gon}(G)\leq 4$, we have $4\leq\mathrm{sn}(G)\leq\operatorname{gon}(G)\leq 4$, so $\mathrm{sn}(G)=\operatorname{gon}(G)=4$.

For some graphs, scramble number is strictly smaller than gonality. In such instances, it is useful to have another upper bound on $\mathrm{sn}(G)$, so that we may rule it out as a tool for reaching a conjectured gonality. To that end we now present the screewidth of a graph, $\operatorname{scw}(G)$, which satisfies $\mathrm{sn}(G)\leq\operatorname{scw}(G)$ [6].

A tree-cut decomposition $\mathcal{T}=(T,\mathcal{X})$ of a graph $G$ is a tree $T$ and a function $\mathcal{X}:V(G)\rightarrow V(T)$, i.e. an assignment (not necessarily one-to-one, not necessarily onto) of the vertices of $G$ to the nodes of some tree $T$. (For clarity, we refer to the vertices and edges of $T$ as its nodes and links, respectively, reserving vertices and edges for $G$.) It is natural to represent a tree-cut decomposition as a thickened drawing of the tree $T$ with the vertices of $G$ inside its nodes, with the edges of $G$ drawn along the unique paths in $T$ between the nodes corresponding to its endpoints. Two examples of tree-cut decompositions of the $4\times 5$ grid graph are illustrated in Figure 5.

Given such a representation of a tree-cut decomposition, we compute several numbers, one for each link and one for each node. For each link $l$ of $T$, we count the number of edges in $G$ passing through $l$. For each node $b$ in $T$, we count the number of vertices of $G$ drawn in $b$, and add the number of “tunneling edges” of $b$, which have neither endpoint in $b$ but still pass through it. The maximum of all these numbers is called the width of $\mathcal{T}$, denoted $w(\mathcal{T})$. In our example, the width of the first is computed as the maximum of $4$, $3$, $3$ (from the links), $12$, $5$, $1$, and $1$ (from the nodes), so its width is $12$; the width of the second decomposition is $4$.

The screewidth of $G$, written $\operatorname{scw}(G)$, is the minimum width of any tree-cut decomposition of $G$. For the $4\times 5$ grid graph, we thus have $\operatorname{scw}(G)\leq 4$; since $4=\mathrm{sn}(G)\leq\operatorname{scw}(G)$, we can also deduce that $\operatorname{scw}(G)=4$. Indeed, without knowing anything about chip-firing games, this could have been used to deduce that $\mathrm{sn}(G)=4$ given that $\mathrm{sn}(G)\geq 4$.

We close this section by recalling several useful bounds on gonality from previous work.

For the next two results, we let $\alpha(G)$ denote the independence number of a graph, which is the largest possible size of an independent set of vertices (i.e., a set of vertices with no two adjacent).

The original proof was for $G$ connected, but can be quickly adapted for arbitrary graphs: choose an independent set $S$ with $|S|=\alpha(G)$, and place a single chip on every vertex besides those in $S$. This divisor has positive rank: to move chips to any vertex $v\in S$, set-fire $\{v\}^{C}$; since $v$ has at least one neighbor in $G$, and all neighbors of $v$ have a chip and exactly one edge connecting them to $v$, this moves at least one chip to $v$ without introducing debt elsewhere.

A king’s graph, denoted $K_{m\times n}$, consists of a grid graph with added diagonals. A $5\times 8$ king’s graph is depicted in Figure 6, with a divisor that we will see has positive rank.

It is natural to ask whether we can push these results further, to find $\operatorname{gon}(K_{m\times n})$ when $3m-2>n$. We present several results to this effect for small values of $m$.

This provides us with $K_{3\times 4}$ as the first instance of a king’s graph whose gonality cannot be computed using scramble number.

We close this section with a discussion of larger knight’s graphs, some of which have unknown gonality. Using a brute-force computation (for instance, using the Chip Firing Interface website [1]), we can determine that $\operatorname{gon}(K_{4\times 4})=10$ and $\operatorname{gon}(K_{4\times 5})=10$. However, it is not possible to prove either result using scramble number. Consider the tree-cut decompositions of $K_{4\times 5}$ in Figure 10. This decomposition has width $8$; since scramble number is non-increasing under taking subgraphs [11, Proposition 4.5], we have $\mathrm{sn}(K_{4\times 4})\leq\mathrm{sn}(K_{4\times 5})\leq\operatorname{scw}(K_{4\times 5})\leq 8$. Thus for $4\times n$ king’s graphs, we cannot use scramble number to compute all gonalities; however, we find computationally that the eventual gonality of $10$ kicks in immediately for the $4\times 4$ graph, in contrast to having lower gonalities for $2\times 2$, $3\times 3$, and $3\times 4$ graphs. In light of this computational result, we pose the following conjecture.

Toroidal king’s graphs add edges to the boundary rows and columns of a usual king’s graph. For $3\leq m\leq n$, this yields an $8$-regular graph, with every vertex having $8$ neighbors. When $m=2$, fewer edges are added, since many of the edges that would be added are already present. We thus split our results across two theorems. First we present a lemma on the independence number of toroidal king’s graphs.

We now present our most general result for toroidal king’s graphs.

We remark that we do not have $\operatorname{gon}(K^{t}_{m\times n})=6m$ for every choice of $m$ and $n$ with $3\leq m\leq n$. For example, $K^{t}_{3\times 3}$ is a complete graph on $9$ vertices, and so has gonality $8$. We can also compute $K^{t}_{3\times 4}$ and $K^{t}_{3\times 5}$ as each vertex has valence strictly larger than half the number of vertices. This implies by Lemma 2.3 that $\operatorname{gon}(K^{t}_{3\times 4})=12-\alpha(K^{t}_{3\times 4})=10$ and $\operatorname{gon}(K^{t}_{3\times 5})=15-\alpha(K^{t}_{3\times 5})=13$. More generally, the upper bound of

gives a better upper bound than $6m$ when $m\leq n\leq 7$.

We now turn to bishop’s graphs. For $n,m\geq 2$, note that $B_{m\times n}$ is disconnected, with exactly two components: one representing white tiles on a chess board, and the other representing black tiles. We denote these two components $B_{m\times n}^{w}$ and $B_{m\times n}^{b}$, and take $B_{m\times n}^{w}$ to contain the vertex $(1,1)$. We remind the reader that for disconnected graphs, gonality can be computed as the sum of the gonalities of the disconnected components, so $\operatorname{gon}(B_{m\times n})=\operatorname{gon}(B_{m\times n}^{w})+\operatorname{gon}(B_{m\times n}^{b})$. Since $B_{2\times in}$ consists of two path graphs, each with gonality $1$ by Lemma 2.1, we have $\operatorname{gon}(B_{2\times n})=2$.

Our general upper bound on $\operatorname{gon}(B_{m\times n})$ with $3\leq m\leq n$ will depend solely on $m$. Before stating this result, we consider $3\times n$, $4\times n$, and $5\times n$ bishop’s graphs. This will allow us to build up a general strategy for the $m\times n$ case.

Recall that $T_{k}=\frac{k(k+1)}{2}$ for $k\geq 1$; by convention, we take $T_{k}=0$ for $k\leq 0$. Note that triangular numbers satisfy the relation $T_{k-1}+\max\{0,k\}=T_{k}$ for any integer $k$.