Book Embeddings of
Chessboard Graphs
Casey J. Hufford
Morehead State University
History of the n-Queens Problem

1848 – Max Bezzel


1850 – Franz Nauck


8-Queens Problem: Can eight queens be placed on an 8x8
board such that no two queens attack one another?
n-Queens Problem: Can n queens be placed on an nxn
board such that no two queens attack one another?
2004 – Chess Variant Pages

Pawn Placement Problem: How many pawns are necessary
to place nine queens on an 8x8 board such that no two
queens can attack one another?
Definition of the Queens Graph



The nxn queens graph Qnxn is the diagram created by
connecting the vertices of two cells on a chessboard with
an edge if a queen can travel from one vertex to the
other in a single turn. (Gripshover 2007)
Qnxn can be broken down into rows, columns, and
diagonals.
A complete graph Kn is a graph on n vertices such that
all possible edges between two vertices exist in the
graph. (Blankenship 2003)
Examples of K4 Graphs
Figure 1: Different representations of a K4
Number of Edges in Qnxn


A complete graph on n vertices has n (n  1) total edges.
2
Qnxn can be broken down into rows, columns, and
diagonals to determine the total number of edges.

Rows:

Columns:

Diagonals:
2
|E| = n (n  1)
2
n 2 (n  1)
|E| =
2
n 1
|E| = n(n-1) + 4 
i 2

Summing the above values
yields:
n 1

|E(Qnxn)| = n(n2-1) + 4  i (i  1)
2
i 2
i (i  1)
2
Broken Down Edges of Q4x4
Figure 2: Q4x4 rows
Figure 3: Q4x4 columns
Figure 4: Q4x4 diagonals
Total Edges of Q4x4
Figure 5: Q4x4
Book Embeddings


A book consists of a set of pages (half-planes)
whose boundaries are identified on a spine
(line). (Blankenship 2003)
To embed a graph in a book linearly order the
vertices in the spine and assign edges to pages
such that:


Each edge is assigned to exactly one page.
No two edges cross in a page.
Book Thickness



The book thickness of a graph G, denoted
BT(G), is the fewest number of pages needed to
embed a graph in a book over all possible vertex
orderings and edge assignments. (Blankenship 2003)
An outerplanar graph can be drawn in a plane
such that no two edges cross and every vertex is
incident with the infinite face.
Useful book thickness results:


BT(G) = 1 if and only if G is outerplanar. (Gripshover 2007)
n
BT(Kn) =   . (Chung, Leighton, Rosenburg 1987),(Blankenship 2003)
2
Book Embedding Examples
4 
Figure 6: Embedding of K4 in  2  , or 2 pages.
 
(Chung, Leighton, Rosenburg 1987)
Figure 7: Embedding of O16 in one page.
(Gripshover 2007)
Past Work:
Queens Graph Upper Bound

MSU undergraduate Kelly Gripshover:

Upper bound involved a combination of graphing
techniques.




Star
Weave
Finagled (manual manipulation)
Focused mainly on the 4x4 queens graph. She found
that BT(Q4x4) ≤ 13. (Gripshover 2007)
Star and Weave Patterns
Figure 8: Star pattern for K5
Figure 9: Weave pattern for Q4x4
Current Work:
Queens Graph Upper Bound

A subgraph H of a graph G has two properties:



The vertex set of H is a subset of the vertex set of G
The edge set of H is a subset of the edge set of G.
In other words, H is obtained from G by a sequence of deleting
edges and vertices of G. Note that if a vertex is deleted, the
edges adjacent to the vertex must also be deleted.
(Bondy, Murty 1981)

Qnxn is a subgraph of the complete graph K n 2.
n 2 
 BT(Qnxn) ≤ BT(Kn 2), which is equivalent to BT(Qnxn) ≤   .
2
Q4x4 Upper Bound
Figure 10: Book embedding of Q4x4 in eight pages.
(Chung, Leighton, Rosenburg 1987)
Definition of
Maximal Outerplanar Graph

A maximal outerplanar graph is an outerplanar
graph such that no edges can be added without
violating the graph’s outerplanarity. (Ku, Wang 2002)
Figure 11: Outerplanar
Figure 12: Maximal outerplanar
Number of Edges in a
Maximal Outerplanar Graph

The number of edges in a maximal outerplanar
graph on n vertices is equal to 2n-3.
Figure 13: n=8, eight adjacent vertices
Figure 14: n=8, five non-adjacent vertices
Past Work:
Queens Graph Lower Bound

BT(G) = 1 if and only if G is outerplanar, so
maximum number of edges embeddable in a
single page is |E(O)|.

|E(Omax)| = 2n2-3 when |V(Omax)| = n2.

Gripshover’s lower bound:

Assumed 2n2-3 edges in every page
Current Work:
Queens Graph Lower Bound

First page has 2n2-3 edges

Every page after first has n2-3 edges

Compare |E(Qnxn)| to maximum number of edges
embeddable in a book with B pages:
n 1
2
n(n -1) + 4  i (i  1) ≤ n2 + B(n2-3)
i 2

2
n 1
i (i  1) 

2
n
(
n

n

1
)

4
Thus, B ≥ 
.


2
i 2


2
n 3




Q4x4 Bound Comparison

Old techniques:


3 ≤ BT(Q4x4) ≤ 13
New techniques:

5 ≤ BT(Q4x4) ≤ 8
Single Pawn Placement


What effect does placing a single pawn on the
board have on the upper and lower bounds?
Two sets of edges are removed:


All edges with the pawn vertex vp as an endpoint.
All edges “crossing over” vp.
Figure 15: Pawn blocking queen movement
Single Pawn Edge Removal
Conjecture: The number of edges
removed depends on the dimensions
of the board, the row number,
and the column number:
(2r+c)n - 3 -
c
r
2
2
(2i-2) -  (2k-3),

i
k
which is equal to
(2r+c)n - 3 - c(c-1) - (r-1)2
where c represents the column number,
r the row, and c ≤ r ≤ n  .
 2 
Figure 18: Fundamental pawn placements (unique pawn placements after any combination
of rotations and reflections) for the 3x3 to 7x7 cases
Single Pawn Lower Bound

The number of edges remaining in Qnxn after single pawn
placement is given by:
n 1
[n(n2-1)


+ 4
i 2
i (i  1)
2
] - [(2r+c)n - 3 - c(c-1) - (r-1)2]
Once again, compare |E(Qnxn(prc))| to the number of
edges in a maximal outerplanar graph on n2 vertices.
Thus, B ≥
n 1
i (i  1)

2
2
n
(
n

n

1
)

4

(
2
r

c
)
n

3

c
(
c

1
)

(
r

1
)



2
i 2


2
n 3




Single Pawn Upper Bound



Upper bound established using complete graphs
Adding a pawn similar (though not equivalent) to
removing vp
Qnxn(prc) is a subgraph of K n  1
2

2

n
 1
BT(Qnxn(prc)) ≤
 2 


Figure 19: Edges remaining after pawn placement
Figure 20: Edges remaining after removing vertex
Summary


The nxn Queensn Graph
Qnxn:
1
n 2 
i
(
i

1
)


2
 n (n  n  1)  4
≤ BT(Qnxn) ≤  



2
2
i 2


n2  3




The nxn Queens Graph After Single Pawn Placement
Qnxn(prc):
n 1
 n (n 2  n  1)  4  i (i  1)  (2r  c )n  3  c (c  1)  (r  1)2 


2
i 2


n2  3




≤ BT(Qnxn(prc)) ≤ n 2  1
 2 


References
F.R.K. Chung, F.T. Leighton, and A.L. Rosenburg, Embedding Graphs in
Books: A Layout Problem with Application to VLSI Design, SIAM J.
Alg. Disc. Meth. 8 (1987), 33-58.
Kelly Gripshover, The Book of Queens, preprint, Morehead State
University, 2007.
J.A. Bondy and U.S.R. Murty, Graph Theory with Applications, 4th ed.
(1981).
Robin Blankenship, Book Embeddings of Graphs, dissertation, Louisiana
State University – Baton Rouge, 2003.
Shan-Chyun Ku and Biing-Feng Wang, An Optimal Simple Parallel
Algorithm for Testing Isomorphism of Maximal Outerplanar Graphs, J.
of Par. and Dist. Com. (2002), 221-227.