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 K6.
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
http://www.rususa.com/
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)
g
C(G)=3
d
a
c
e
b
f
v
a
c(v)
3
b
c
d
e
1
2
2
2
f
g
2
2
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 K2.
Therefore c(v) is equal to
the number of edges
incident on that vertex.
References
http://npluskqueens.info
R.D. Chatham, G.H. Fricke, and R.D. Skaggs, The Queens
Separation Problem, Util. Math. 69 (2006), 129-141
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): 125-127
Harless, Joe, Transit Graphs: Separation, Domination, and
Other Parameters, preprint, Morehead State University,
2007