Chessboard complex

A chessboard complex is a particular kind of abstract simplicial complex, which has various applications in topological graph theory and algebraic topology.^[1][2] 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 ${\displaystyle \Delta _{m,n}}$ is the abstract simplicial complex with vertex set ${\displaystyle [m]\times [n]}$ that contains all subsets S such that, if ${\displaystyle (i_{1},j_{1})}$ and ${\displaystyle (i_{2},j_{2})}$ are two distinct elements of S, then both ${\displaystyle i_{1}\neq i_{2}}$ and ${\displaystyle j_{1}\neq j_{2}}$. 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 ${\displaystyle (D_{m})_{\Delta (2)}^{*n}}$.^[3]: 176

Another definition is the set of all matchings in the complete bipartite graph ${\displaystyle K_{m,n}}$.^[1]

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:^[4]

• 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 ${\displaystyle \Delta _{m,n}}$ contains ${\displaystyle \min(m,n)}$ elements. Therefore, the dimension of ${\displaystyle \Delta _{m,n}}$ is ${\displaystyle \min(m,n)-1}$.

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

The Betti numbers ${\displaystyle b_{r-1}}$ of chessboard complexes are zero if and only if ${\displaystyle (m-r)(n-r)>r}$.^[5]: 200 The eigenvalues of the combinatorial Laplacians of the chessboard complex are integers.^[5]: 193

The chessboard complex is ${\displaystyle (\nu _{m,n}-1)}$-connected, where ${\displaystyle \nu _{m,n}:=\min\{m,n,\lfloor {\frac {m+n+1}{3}}\rfloor \}}$.^[6]: 527 The homology group ${\displaystyle H_{\nu _{m,n}}(M_{m,n})}$ is a 3-group of exponent at most 9, and is known to be exactly the cyclic group on 3 elements when ${\displaystyle m+n\equiv 1{\pmod {3}}}$.^{[6]: 543–555}

The ${\displaystyle (\lfloor {\frac {n+m+1}{3}}\rfloor -1)}$-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 ${\displaystyle n\geq 2m-1}$.^[7]: 3 As a corollary, any position of k rooks on a m-by-n chessboard, where ${\displaystyle k\leq \lfloor {\frac {m+n+1}{3}}\rfloor }$, can be transformed into any other position using at most ${\displaystyle mn-k}$ single-rook moves (where each intermediate position is also not rook-taking).^[7]: 3

Generalizations

The complex ${\displaystyle \Delta _{n_{1},\ldots ,n_{k}}}$ 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 ${\displaystyle (\nu -2)}$-connected, for ${\displaystyle \nu$ :=\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 \}} ^[1]: 33