Chessboard complex

Short description: Mathematical object in topological graph theory

A chessboard complex is a particular kind of abstract simplicial complex, which has various applications in topological graph theory and algebraic topology. Informally, the (m, n)-chessboard complex contains all sets of positions on an m-by-n chessboard, where rooks can be placed without attacking each other. Equivalently, it is the matching complex of the (m, n)-complete bipartite graph, or the independence complex of the m-by-n rook's graph.

Definitions

For any two positive integers m and n, the (m, n)-chessboard complex [math]\displaystyle{ \Delta_{m,n} }[/math] is the abstract simplicial complex with vertex set [math]\displaystyle{ [m]\times [n] }[/math] that contains all subsets S such that, if [math]\displaystyle{ (i_1,j_1) }[/math] and [math]\displaystyle{ (i_2,j_2) }[/math] are two distinct elements of S, then both [math]\displaystyle{ i_1\neq i_2 }[/math] and [math]\displaystyle{ j_1\neq j_2 }[/math]. The vertex set can be viewed as a two-dimensional grid (a "chessboard"), and the complex contains all subsets S that do not contain two cells in the same row or in the same column. In other words, all subset S such that rooks can be placed on them without taking each other.

The chessboard complex can also be defined succinctly using deleted join. Let D_m be a set of m discrete points. Then the chessboard complex is the n-fold 2-wise deleted join of D_m, denoted by [math]\displaystyle{ (D_m)^{*n}_{\Delta(2)} }[/math].

Another definition is the set of all matchings in the complete bipartite graph [math]\displaystyle{ K_{m,n} }[/math].

Examples

In any (m,n)-chessboard complex, the neighborhood of each vertex has the structure of a (m − 1,n − 1)-chessboard complex. In terms of chess rooks, placing one rook on the board eliminates the remaining squares in the same row and column, leaving a smaller set of rows and columns where additional rooks can be placed. This allows the topological structure of a chessboard to be studied hierarchically, based on its lower-dimensional structures. An example of this occurs with the (4,5)-chessboard complex, and the (3,4)- and (2,3)-chessboard complexes within it:

• The (2,3)-chessboard complex is a hexagon, consisting of six vertices (the six squares of the chessboard) connected by six edges (pairs of non-attacking squares).
• The (3,4)-chessboard complex is a triangulation of a torus, with 24 triangles (triples of non-attacking squares), 36 edges, and 12 vertices. Six triangles meet at each vertex, in the same hexagonal pattern as the (2,3)-chessboard complex.
• The (4,5)-chessboard complex forms a three-dimensional pseudomanifold: in the neighborhood of each vertex, 24 tetrahedra meet, in the pattern of a torus, instead of the spherical pattern that would be required of a manifold. If the vertices are removed from this space, the result can be given a geometric structure as a cusped hyperbolic 3-manifold, topologically equivalent to the link complement of a 20-component link.

Properties

Every facet of [math]\displaystyle{ \Delta_{m,n} }[/math] contains [math]\displaystyle{ \min(m,n) }[/math] elements. Therefore, the dimension of [math]\displaystyle{ \Delta_{m,n} }[/math] is [math]\displaystyle{ \min(m,n)-1 }[/math].

The homotopical connectivity of the chessboard complex is at least [math]\displaystyle{ \min\left(m, n, \frac{m+n+1}{3}\right)-2 }[/math] (so [math]\displaystyle{ \eta \geq \min\left(m, n, \frac{m+n+1}{3}\right) }[/math]).

The Betti numbers [math]\displaystyle{ b_{r - 1} }[/math] of chessboard complexes are zero if and only if [math]\displaystyle{ (m - r)(n - r) \gt r }[/math]. The eigenvalues of the combinatorial Laplacians of the chessboard complex are integers.

The chessboard complex is [math]\displaystyle{ (\nu_{m, n} - 1) }[/math]-connected, where [math]\displaystyle{ \nu_{m, n} := \min\{m, n, \lfloor\frac{m + n + 1}{3}\rfloor \} }[/math]. The homology group [math]\displaystyle{ H_{\nu_{m, n}}(M_{m, n}) }[/math] is a 3-group of exponent at most 9, and is known to be exactly the cyclic group on 3 elements when [math]\displaystyle{ m + n \equiv 1\pmod{3} }[/math].

The [math]\displaystyle{ (\lfloor\frac{n + m + 1}{3}\rfloor - 1) }[/math]-skeleton of chessboard complex is vertex decomposable in the sense of Provan and Billera (and thus shellable), and the entire complex is vertex decomposable if [math]\displaystyle{ n\geq 2m - 1 }[/math]. As a corollary, any position of k rooks on a m-by-n chessboard, where [math]\displaystyle{ k\leq\lfloor\frac{m + n + 1}{3}\rfloor }[/math], can be transformed into any other position using at most [math]\displaystyle{ mn - k }[/math] single-rook moves (where each intermediate position is also not rook-taking).

Generalizations

The complex [math]\displaystyle{ \Delta_{n_1,\ldots,n_k} }[/math] is a "chessboard complex" defined for a k-dimensional chessboard. Equivalently, it is the set of matchings in a complete k-partite hypergraph. This complex is at least [math]\displaystyle{ (\nu - 2) }[/math]-connected, for [math]\displaystyle{ \nu := \min\{n_1, \lfloor\frac{n_1 + n_2 + 1}{3}\rfloor, \dots, \lfloor\frac{n_1 + n_2 + \dots + n_k + 1}{2k + 1}\rfloor\} }[/math]