The Equivalence Number and Transit Graphs for Chessboard Graphs B. Nicholas Wahle Morehead State University

Graphs A graph is a set of points called vertices with unordered pairs of vertices called edges.

Paths A path is a subset of the vertices such that there is an edge connecting one vertex to the next. The vertices and edges in red form a P4.

Complete Graph A complete graph, is a graph in which all vertices are adjacent to every other vertex by an edge. This graph forms a K 6.

Cliques A clique is a subset of the vertices such that the subset forms a complete graph. The vertices and edges in red form a clique of order 5.

Independence An independent set of vertices in a graph is a set such that none of the vertices are joined by an edge. The independence number of the graph is the largest number of independent vertices that can be found. The vertices in red form an independent set.

N-Queens Problem The original Queens problem asked if eight queens could be placed on a standard 8x8 chessboard such that no two queens attack each other. (Bezzel, 1848) It was later generalized as N queens being placed on a NxN chessboard for N larger than 4. (Nauck, 1850)

N+k Queens Problem The NxN board could not contain more than N queens, since a queen can attack any space in its row. More queens can be added to the board by placing pawns to block their attacks. Given a large enough N, it has been shown that N+k queens can be placed on an NxN board with k pawns separating them. (Chatham, et al 2006)

Queens Graph A queens graph is a graph where each square of a chessboard is represented by a vertex in the graph. The graph has an edge connecting two vertices if a queen can move from one square to the other in a legal move.

Pieces

Transit Graphs Let F be a family of graphs on the same vertex set, V. The transit graph of F is the graph on V such that ab is an edge if and only if there is a path from a to b in one of the graphs of F. The elements of F are called routes. The equivalence number of a graph, eq(G), is the minimum number of routes required to construct the graph G.

Another Look The routes are similar to a subway map or a road map. The maps show you where you can go without having to change roads or subway trains. Image from

A Minimal Example

A Minimum Example So eq(G)=3 for this graph.

Covering a Vertex Given a vertex v, define c(v) to be the minimum number of cliques required to cover all edges incident with vertex v. Define C(G) to be the maximum c(v) of all the vertices of the graph G.

Finding C(G) C(G)=3 c a d e g f b vc(v)c(v) a3 b1 c2 d2 e2 f2 g2

Bounds on Equivalence We have shown that C(G) ≤ eq(G) and conjectured that eq(G) ≤ C(G) + 1 Since C(G) cliques are required to cover at least one vertex and at most one clique containing that vertex can be represented in a route, C(G) ≤ eq(G). Currently there is not a proof for eq(G) ≤ C(G)+1 nor has it been disproved.

Why?

Remove a Vertex

Putting Together the Pieces

Other Chess Pieces For the queens graph, the equivalence number is 4, for a 4x4 or larger board. The rooks graph has an equivalence number of 2, for 2x2 board or larger. For a 3x3 board or larger, the bishops graph has an equivalence number of 2. A knights graph has an equivalence number of 8, for a board 5x5 or larger.

Knights Graph The knights graph does not allow for a clique larger than a K 2. Therefore c(v) is equal to the number of edges incident on that vertex.

References R.D. Chatham, G.H. Fricke, and R.D. Skaggs, The Queens Separation Problem, Util. Math. 69 (2006), Chatham, Douglas, et al, Independence and Domination on Chessboard Graphs, preprint, Morehead State University, 2006 Frankl, Peter, Covering Graphs by Equivalence Relations. Annals of Discrete Mathematics 12 (1982): Harless, Joe, Transit Graphs: Separation, Domination, and Other Parameters, preprint, Morehead State University, 2007