On the Domination Chain of m by n Chess Graphs

Transcription

1 Murray State's Digital Coons Murray State Theses and Dissertations Graduate School 018 On the Doination Chain of by n Chess Graphs Kathleen Johnson Follow this and additional works at: Part of the Discrete Matheatics and Cobinatorics Coons Recoended Citation Johnson, Kathleen, "On the Doination Chain of by n Chess Graphs" (018). Murray State Theses and Dissertations This Thesis is brought to you for free and open access by the Graduate School at Murray State's Digital Coons. It has been accepted for inclusion in Murray State Theses and Dissertations by an authorized adinistrator of Murray State's Digital Coons. For ore inforation, please contact su.digitalcoons@urraystate.edu.

2 On the Doination Chain of n Chess Graphs A Thesis Presented to the Faculty of the Departent of Matheatics and Statistics Murray State University Murray, Kentucky In Partial Fulfillent of the Requireents for the Degree of Master of Science by Kathleen G. Johnson May, 018

3 On the Doination Chain of n Chess Graphs DATE APPROVED: Dr. Elizabeth Donovan, Thesis Advisor Dr. Robert Donnelly, Thesis Coittee Dr. Scott Lewis, Thesis Coittee Dr. Kevin Revell, Graduate Coordinator, College of Sci., Eng., and Tech. Dr. Stephen Cobb, Dean, College of Sci., Eng., and Tech. Dr. Robert Pervine, University Graduate Coordinator Dr. Mark Arant, Provost

4 Acknowledgeents I would like to thank y thesis advisor Dr. Elizabeth Donovan for pushing e to to do ore for y thesis than I thought possible. I also extend y thanks to the ebers of y thesis coittee Dr. Robert Donnelly and Dr. Scott Lewis for their advice and support. I would also like to thank the Murray State University Math Departent for helping e through this process. To y fellow graduate students your encourageent has ade this endeavor anageable. Many thanks to Ada Bender for his brilliantly siple idea that black and white squares are different colors. Finally, I would like to thank y parents: y o for laying the foundation of y interest in this topic by teaching e how to play chess at a young age and y dad for always being there for e. iii

5 Abstract The gae of chess has fascinated people for hundreds of years in any countries across the globe. Chess is one of the ost challenging and well-studied gaes of skill. The underlying aspects of chess give rise to any classically studied puzzles on the chessboard. Graph theoretic analysis has been used to study nuerous chess-related questions. We relay results on various non-attacking packings and coverings for the rook, bishop, king, queen, and knight on square and nonsquare chessboards. These results lay the foundation for our work with the n Bishop graph. We exaine the role of the bishop in the non-attacking packing proble as well as the covering proble for oblong chessboards and present constructions for the doination nuber and independence nuber of such graphs. iv

6 Contents 1 Introduction 1 Chess Gaes 3.1 A Brief History of Chess Chessboard Layout Chess-piece Moves Chess Variants Variations on Modern Chess Graph Theory Definitions Basic Terinology Packing and Covering Paraeters Chess Graphs Known Results Rook Bishop King Queen Knight v

7 5 New Results Bishop Moveent on an n Graph Bishop Graph Definitions Independence of the n Bishop Graph Doination of the n Bishop Graph References 57 vi

8 1 Chapter 1 Introduction The strongest chess players were often considered worldwide chapions as far back as the Middle Ages. Since chess is a gae of strategy, atheaticians have endeavored to build algoriths that would allow anyone or any achine to play a successful gae against such a chapion since at least the 18 th century. One faous outcoe of this abition is the Turk as told by Schaffer [17]. This autoaton built in 1770 could successfully play chess against a huan player and coplete the atheatical chess puzzle known as the Knight s Tour. The Turk itself went on exhibition across Europe and North Aerica playing against strong chess players for alost 84 years. The Turk was confired to be a fake that is, a person hiding inside the achine playing the gae, by the son of the last owner Mitchell [15], in The fact that this was not revealed for alost one hundred years speaks to our desire to find a forulation that can do anything we want it to do. The coputer chess chapion abition cae to fruition in 1997 when chess-playing coputer Deep Blue beat a current world chess chapion. While any probles involving chess have been solved the gae still has nuerous puzzles for which we have yet to produce a solution. The algorithic nature of chess lends itself naturally to atheatically-inclined puzzles. In particular, the oveent

9 of the pieces have inspired nuerous probles, any of which are still unsolved today. Even as older probles are solved, new ones take their place as we iagine chess on different types of chessboards and enact new ethods of play. In our efforts to solve these probles in the present, we have atheatical tools such as graph theory to describe and discover solutions to existing probles. Our purpose is to exaine and provide new inforation involving classic graph theoretic chess probles. We exaine the covering proble, non-attacking covering proble, and packing proble, aong others. These probles are applied to various chess pieces on various shaped chessboards. The chess pieces studied are five of the six classic pieces: the queen, the king, rooks, bishops, and knights. While the above entioned probles have been studied at length for n n boards not uch is known for n boards. We will take this avenue in our research, focusing on the packing and covering probles for the bishop. In Chapter we lay the groundwork for our use of chess, including the layout of the board and the oveent of the five pieces under consideration. In Chapter 3 we begin our discussion of graph theory, drawing together the necessary definitions with our perspective of chess graph and the doination chain. In Chapter 4 we exaine known results for the doination chain for the rook, bishop, king, queen, and knight graphs. Chapter 5 contains our findings fro exaining the oveent of the bishop on a rectangular board, right triangular board, and trapezoidal board.

10 3 Chapter Chess Gaes.1 A Brief History of Chess Chess is an ancient gae that began in 6 th century India and spread throughout the world as it evolved into the for we know today [16]. Chess is a war gae played between two players. The two players begin the gae with the sae pieces and settings so that the players ust use tactics and strategy to win. Modern chess developed during the Middle Ages. As chess becae ore popular in Europe, so did chess-based puzzles. Watkins [18] writes that the earliest chessboard puzzle he knows of is Guarini s Proble fro 151. Several faous atheaticians have worked on various chess puzzles, including Leonhard Euler and Carl Friedrich Gauss who worked on The Knight s Tour and Eight Queens Proble, respectively. Queens Doination is another popular proble. Probles about the queen piece are well-studied and have been expanded into the other chess pieces. Probles on an n n board have been thoroughly considered. We exaine soe less well investigated probles on the n board. Matheaticians and coputer scientists have used coputer algoriths to resolve chess probles since the id-twentieth century.

11 .. Chessboard Layout 4. Chessboard Layout A standard chessboard is the 8 8 checkerboard. This board consists of eight coluns and eight rows ade up of an equal nuber of alternating black and white squares. There are 3 chess pieces to start a standard gae of chess, 16 white pieces and 16 black pieces, covering half of the standard 64 squares. Of each color set, there is one queen q, one king k, two bishops b, two knights n, two rooks r, and eight pawns p. At the beginning of a standard gae, each player lays out his/her pieces as shown in Figure..1. rblkans opopopop 0Z0Z0Z0Z Z0Z0Z0Z0 0Z0Z0Z0Z Z0Z0Z0Z0 POPOPOPO SNAQJBMR Figure..1: The layout of a standard gae of chess. Note that we will follow the convention of a black square in the lower left corner of the board..3 Chess-piece Moves While the classic gae of chess raises soe interesting atheatical questions, our focus will be ore directed toward the oveent of the pieces. Each piece has a distinct set of oves that it can ake. A rook can ove any nuber of squares either horizontally or vertically, while a bishop can ove any distance diagonally. A knight can ove on a chessboard by going two squares in any horizontal or vertical

12 .3. Chess-piece Moves 5 direction, and then turning either left or right one ore square [18], thus oving in an L. The king s ove is ore restrictive: it ay ove only one square horizontally, vertically, or diagonally. The queen has the ost control over the board as it can ove any nuber of squares horizontally, vertically, or diagonally, thus aking it a cobination of bishop and rook oveent. We will not discuss the oveent options for the pawn as it is dependent on the current layout of the board. The above oveent of these pieces can be see in Figure.3.1. Notice in Figure.3.1b that a bishop has a restriction on its location: it can only ever ove to squares of its original starting color. That is, a bishop on white can only control white squares and a bishop on black can only control black squares. Z0Z0Z 0Z0Z0 Z0s0Z 0Z0Z0 Z0Z0Z Z0Z0Z 0Z0Z0 Z0a0Z 0Z0Z0 Z0Z0Z Z0Z0Z 0Z0Z0 Z0j0Z 0Z0Z0 Z0Z0Z (a) Rook (b) Bishop (c) King Z0Z0Z 0Z0Z0 Z0l0Z 0Z0Z0 Z0Z0Z Z0Z0Z 0Z0Z0 Z00Z 0Z0Z0 Z0Z0Z (d) Queen (e) Knight Figure.3.1: Five chess pieces and their oves. Fro Figures.3.1c and.3.1e we can see both the king and the knight has the

13 .4. Chess Variants 6 possibility to attack, or control, at ost eight squares, while the liiting factor for the attacking options of a rook, bishop, or queen (Figures.3.1a,.3.1b, and.3.1d)are deterined by the size of the board. Notice that the queen can attack ore squares than the rook or bishop alone, since its oveent is a cobination of both the bishop and the rook otions..4 Chess Variants As chess evolved in different countries, it took a variety of different fors. It began as the Indian gae Chaturanga and now several different versions of the gae exist in Asia. A particularly notable chess-variant is Shogi fro Japan. Shogi is played on a 9 9 board with five of the classic chess pieces (rook, bishop, king, knight, pawn) as well as soe additional pieces (gold and silver generals, and lance). A ajor difference is that certain pieces can be prooted and acquire additional oveent. The rook piece advances to becoe the dragon king and the bishop piece can becoe the dragon horse. These advanced otions ake these pieces the ost siilar to the classic queen piece than any of the other five standard pieces. We will see later in Chapter 4 these pieces have been studied as part of further research into Queens Doination..5 Variations on Modern Chess Modern chess can played on different surfaces as well. Besides a rectangular board, including those that are square, chess oveents have been analyzed on a torus, various 3D-boards, triangular boards [18], boards with hexagons in the place of squares [6], just to nae a few. Each of these board shapes has given rise to different results of the ore classically studies chess probles.

14 7 Chapter 3 Graph Theory Definitions 3.1 Basic Terinology Various graph theory paraeters have been applied to the chess graphs constructed by the oveents of each of the entioned five chess pieces. To gain insight into these constructions, we begin with the necessary graph theory definitions. Definition A graph is a pair G = (V, E) consisting of a vertex set V (G) (or V when the graph is understood) together with an edge set E(G) (or E), which is coprised of -eleent subsets of V, the endpoints of an edge. Two vertices that for an edge are adjacent vertices and so they are neighbors. The order of a graph G denoted n(g) refers to the nuber of vertices and the size of the graph e(g) is the nuber of edges. The degree of a vertex v is the nuber of edges with v as at least one of its endpoints. See Figure for an exaple of a graph and these paraeters. When working with a graph we often need to focus on a certain subset of the graph. Such a subset is ore forally set out in the following definitions: Definition A subgraph has vertices and edges belonging to G.

15 3.1. Basic Terinology 8 e u b z c v d x y w a Figure 3.1.1: Graph G with vertex set V (G) = {a, b, c, d, e} and edge set E(G) = {u, v, w, x, y, z}. n(g) = 5 and e(g) = 6 with the vertices a, b, and c having the largest degree of 3. a and b are adjacent and are said to be neighbors as they are the endpoints of edge x. Definition An induced subgraph G[A] has vertex set A V (G) obtained by taking A and all edges of G having both endpoints in A. Figure 3.1. gives an exaple of a graph with a subgraph and induced subgraph. h g f i a b c e d g f i a b c e d h f a b c e d (a) A graph G with vertex set V = {a, b, c, d, e, f, g, h, i}. (b) A subgraph H of G. (c) An induced subgraph G[A] with vertex set A = {a, b, c, d, e, f, h}. Figure 3.1.: A graph G with induced subgraph H and induced subgraph G[A]. Definition A coponent is a axial connected subgraph. Definition A connected graph G has a u, v-path for each set of distinct vertices u, v. That is every vertex u, has a path to every vertex v, u, v V (G), u v.

16 3.1. Basic Terinology 9 Definition A bipartite graph G has vertex sets X and Y such that each edge in G has one vertex in X and one vertex in Y. Observe an exaple a bipartite graph that is connected in Figure 3.1.3a and of a disconnected graph in Figure 3.1.3b The connected graph is coprised of one coponent while the disconnected graph is coposed of three coponents, one of which is a singleton vertex. (a) A connected graph that is bipartite. (b) A disconnected graph with 3 coponents. Figure 3.1.3: An exaple of a bipartite graph that is connected and a disconnected graph. Of particular iportance in this research is the claw graph. The claw graph, ore forally known as K 1,3, consists of one vertex, often depicted in the iddle of the graph, which is adjacent to the reaining three vertices in the graph. This graph is depicted in Figure Definition A graph is considered claw-free if it does not contain the claw graph as an induced subgraph. Note that the graph H given in Figure 3.1.b is not claw-free as the induced subgraph H[S] with S = {a, b, g, c} is a claw, while the graph G[A] in Figure 3.1.c is, in fact, claw-free.

17 3.. Packing and Covering Paraeters 10 Figure 3.1.4: The claw graph. 3. Packing and Covering Paraeters Our research is focused on various packing and covering paraeters of a graph. Specifically, these covering and packing probles will often be focused on the doination or independence of vertices, causing us to iniize or axiize our vertex selection, respectively. As a note the or between doination and independence is not exclusive we will also be considering vertex independent doination which ust satisfy the requireents of both paraeters. Fro there, we will consider one other related paraeter, the irredundance nuber, and connect these ideas with the doination chain. Consider the following question: deterine a set of vertices so that every vertex is adjacent to at least one vertex in this set. This idea of a covering proble lends itself to be viewed as a doination paraeter. Doination probles are well-studied in graph theory due to their practical applications. For exaple, Watkins [18] draws a connection between chess and doination in that chess began as a gae of war. Thus it is of no surprise that the idea of doination and chess go together. Definition A doinating set is a set S V such that every vertex outside S has a neighbor in S. That is, every vertex not in S is adjacent to a vertex in S. Clearly we can ake this set as large as we wish, up to including all of the vertices in our graph. Thus, the challenge becoes deterining how sall we can ake S: a inial doinating set that has the fewest vertices needed to doinate the graph.

18 3.. Packing and Covering Paraeters 11 Definition 3... The doination nuber of a graph γ(g) is the iniu cardinality of a inial doinating set of vertices. A doinating set of size γ(g) is known as a iniu doinating set. Definition The upper doination nuber of a graph Γ(G) is the axiu cardinality of a inial doinating set. A doinating set of size Γ(G) is known as a axiu doinating set. i h g f i h g f a b j c d e a b j c d e (a) Doinating set {b, d, g} in blue. (b) Doinating set {c, e, f, h, i} in blue. Figure 3..1: Doinating sets for a graph L. In Figure 3..1a we see that a iniu doinating set is {b, d, g}, aking the doination nuber γ(l) = 3. There is a axiu doinating set {b, e, f, i, j}, shown in blue in Figure 3..1b, giving the upper doination nuber Γ(L) = 5. Given a graph, how any vertices can we choose so that none of the are neighbors? Let us forally define this idea. Definition ([0]) An independent set is a set of pairwise nonadjacent vertices. For this paraeter it is easy to see that we can ake our set of vertices as sall as we wish we could select the epty set of vertices and still satisfy the above definition. Therefore, we will seek out the largest possible set of vertices that still satisfies the given condition: a axial independent set has a set of vertices such that the addition of another vertex would cause the set to be not independent. Definition ([0]) The independence nuber of a graph α(g) is the axiu size of an independent set of vertices.

19 3.. Packing and Covering Paraeters 1 e c a (a) A graph M with independent set {b, d, e} in blue. d b f e c a (b) A graph N with independent doinating set {c, f} in blue. d f b Figure 3..: An exaple of independent and independent doinating sets. We ay require a set of vertices to be both doinant and independent. This idea leads to the next paraeter. Definition ([13]) The independent doination nuber i(g) of a graph is the iniu cardinality of an independent doinating set. Gross, et al. [10] defined the class of doination perfect graphs through its induced subgraphs, while Goddard found the sae connection through the exclusion of the claw graph. Definition [10] A graph G is doination perfect if for every induced subgraph H, γ(h) = i(h). Theore [9] If a graph G is claw-free graph, then γ(g) = i(g). Corollary [9] If a graph G is claw-free graph, then G is doination perfect. Several failies of graphs have their own properties. For instance the cycle graph faily C n, in which C 6 is shown in Figure (3..a), has independence nuber α(c n ) = n where n denotes the order of Cn. In Figure (3..1a), we see that our doinating set is not independent, and so our independent doination nuber i(l) ust be greater than or equal to our doination nuber γ(l). Definition ([13]) A set S of vertices in a graph is called an irredundant set if for each vertex v S either v itself is not adjacent to any other vertex in S or else

20 3.. Packing and Covering Paraeters 13 there is at least one vertex u / S such that u is adjacent to v but to no other vertex in S. That is, v has a private neighbor or is a private neighbor itself. The difficulty in creating an irredundant set coes when aking the set as large as possible. Optiizing this packing proble can be be done in two different ways, as set out in the following definitions. Definition ([13]) The irredundance nuber of a graph ir(g) is the iniu cardinality of a axial irredundant set of vertices. Definition ([13]) The upper irredundance nuber of a graph IR(G) is the axiu cardinality of an irredundant set of vertices. (a) An irredundant set of size 3. Note that this set is not doinating. (b) An irredundant set of size 5. Figure 3..3: Irredundance graphs for W. Note that ir(w ) = 3 as seen on the left while IR(W ) = 5, shown on the right. As stated in Chapter 1, we will exaine the covering proble, non-attacking covering proble, and packing proble for various chess pieces. In the context of graph theory we now refer to the as doination, independent doination, and independence probles. We study these paraeters as well as the other covering and packing paraeters: irredundance, upper doination and upper irredundance. The following holds true for any graph G and is known as a doination chain according to Gross, Yellen, and Zhang [10]. Theore ([10]) ir(g) γ(g) i(g) α(g) Γ(G) IR(G)

21 3.3. Chess Graphs 14 The first three values in the chain iniize sets and the last three axiize sets. For certain classifictions of graphs, these three iniization paraeters, ir(g), γ(g), and i(g) tend to be equal and the sae can be said for the axiized values α(g), Γ(G), and IR(G). In fact, for a bipartite graph, Cockayne et al. [4] proved that the axiized values are the sae. Theore ([4]) If G is a bipartite graph, then α(g) = Γ(G) = IR(G). Watkins [18] takes an intuitive approach to the relationship between the doination chain paraeters. For the iniizing nubers, as we add additional constraints to finding a iniu set of vertices, such a set can only increase in size. ir(g) γ(g) since a doinating set is an irredundant set that also doinates, and, hence, the extra conditions can only ake a required set larger. Transitioning to consider the second and third paraeters, we find that adding the requireent of independence yields γ(g) i(g) siilarly. For our axiization concerning the reaining three paraeters, we start with the restriction that our vertices ust be independent, liiting the axiu size of our set. Therefore, α(g) Γ(G) as the independence restriction is reoved. Siilarly, Γ(G) IR(G) since the doinating restriction is reoved when considering the upper irredundance nuber. Doination and independence nubers are the ost popular of the six to study as they have wide-ranging practical applications. Aong these six paraters the irredundance nubers are the ost difficult to study and, subsequently, have the least nuber of known results. 3.3 Chess Graphs We will now cobine the ideas of chess piece oveents and graph theory. As noted in [18], the chessboard and otion of the pieces are represented very well using graphs. Each chess piece akes its own graph using a vertex to represent a single square on

22 3.3. Chess Graphs 15 the board and edges to represent the oveent of the piece fro a certain square to other allowable squares. In this way we can ake graphs for each piece that look siilar to the figures in Section.3. Notice that for each chess piece, their graphs will all have the sae order when we fix the board size. The difference between the figures given before and these graphs is that the graph shows every possible ove fro every possible square on the board. This will cause the graph to have a large aount of edges, which often akes its visualization difficult. To siplify the graph, we will use the convention of varied line thicknesses: when all the edges are drawn, thinner edges will be used; when thicker edges are drawn vertices are adjacent if they lie on the sae line (vertical, horizontal or diagonal). We begin with the rook graph. The Rook graph is the ost straightforward graph, consisting of horizontal and vertical edges. This is due to the otion of the rook, which allows the piece to ove any nuber of spaces along its current row or colun on the board. (a) Rook graph (b) Siplified Rook Graph Figure 3.3.1: The 3 3 Rook board represented as a graph and a siplified graph. A 3 3 Bishop graph has only edges representing diagonal otion as seen earlier in Section.3. A Bishop graph on ore than one vertex is always disconnected and, specifically, a Bishop graph on a single row or colun is coprised only of isolated vertices. For graphs on ore than one row or colun, the black and white squares each for their own coponents of the graph since a bishop oving diagonally can only travel across sae-colored squares. Because of this separability, we will often only consider one coponent at a tie in our analysis.

23 3.3. Chess Graphs 16 (a) Bishop graph (b) Siplified Bishop Graph Figure 3.3.: The 3 3 Bishop board represented as a graph and a siplified graph. Recall that the oveent of the king allows the piece to ove only one square in any of eight possible directions. Since the piece is restricted to oving only one square at a tie, its siplified graph is identical to the original graph. (a) King graph (b) Siplified King Graph Figure 3.3.3: This is the 3 3 King board represented as a graph and then siplified. Note that since the king can ove only one space in any direction no siplified versions of edged can be used. Aong all standard chess pieces, the queen has the ost freedo in its oveent. Because of this, its graph will be the ost coplicated. The queen cobines the otions of a rook and a bishop. Also, the Rook, Bishop, and King graphs are all subgraphs of the Queen graph for a given board size. We can see in Figures 3.3.1, 3.3., and that the pieces that can ove ore than one square in any direction have ore coplicated graphs and this coplication will grow faster as the size of the board increases. Hence, it is easier to analyze the siplified graphs, reebering that the rook, bishop, and queen can ove any distance along a straight path. The Knight graph is already siplified as it cannot ove ore than one L

24 3.3. Chess Graphs 17 (a) Queen graph (b) Siplified Queen Graph Figure 3.3.4: The 3 3 Queen board represented as a graph and a siplified graph. shape at a tie. The Knight graph can be disconnected depending on the nuber of squares in the original board. In fact, it is known to always be disconnected if the board contains either one or two rows or coluns. The 3 3 exaple given in Figure has an isolated vertex in the center, since a knight starting at this position cannot travel two squares out in any one direction and, thus, cannot coplete an L otion. The Knight graph is also bipartite since in its L otion, it oves two squares over, to the sae colored square and then over one ore either left or right, both of which ust be opposite colored squares. Thus, the Knight graph has two partitions: the black squares and the white squares. (a) Knight graph (b) Siplified Knight Graph Figure 3.3.5: This is the 3 3 Knight board represented as a graph and then siplified.

25 18 Chapter 4 Known Results In the following sections we will discuss soe of the known doination chain results for each of the five ajor chess pieces. This short survey encopasses known results fro chess piece graphs on both n n and n boards. Soe of the earliest work was done by Yaglo and Yaglo [] for n n King, Bishop, and Rook graphs. Two previous n n surveys by Fricke et al. [8] and Haynes, Hedetniei, and Slater [1] have filled in the gaps in the Bishop and Rook graphs such that the whole of the six-paraeter doination chain is known for each of these pieces. These surveys also provide additional results for the other chess pieces. When paraeters are not known we give bounds, especially in the case of the Queen graph. Probles solved by Yaglo and Yaglo [] are the proofs of the theores stated in Fricke et al. [8] and Haynes, Hedetniei, and Slater [1]. 4.1 Rook As noted above, every paraeter in our doination chain is known for R(n, n) = R n. Fro the otion of the rook piece it is easy to see that n rooks are needed to cover a n n Rook graph as seen in Figure 4.1.1a.

26 4.1. Rook 19 Theore ([8]) For n 31, ir(r n ) = n. Theore ([], [8]) For n 1, γ(r n ) = i(r n ) = α(r n ) = n. One siple construction for finding a doinating set is to select a set of vertices corresponding to a single row or colun on the given square chessboard (Figure 4.1.1a); to find an independent doinating set, we, for exaple, ay instead choose those vertices on any one of the two ain diagonals (Figure 4.1.1b). R n extends naturally into R(, n) for soe results as the Rook has unliited horizontal and vertical oveent, so extending the board by one row or colun still ensures it is covered. For exaple, we can still find a doinating set by selecting either an entire or colun fro the board. However, to iniize this paraeter, as is needed to deterine γ(r(, n)), we would select the saller of the two possibilities: if there are ore coluns than rows, select a set of vertices corresponding to a single colun and vice versa if the nuber of row is greater than (or equal to) the nuber of coluns. Siilar to Theore 4.1., equality aong the three iniizing paraeters in the doination chain are all equal for R(, n). See Figure 4.1. for an exaple on a 4 3 graph. Corollary [14] For n 1, γ(r(, n)) = i(r(, n)) = α(r(, n)) = in{, n}. The straightforward nature of the Rook graph yields that every paraeter for R n has value n except the upper irredundance nuber IR(R n ). We see that IR(R n ) = n up to n = 4, but quickly grows past this value as n gets large. An exaple of the general construction for IR(R n ) for n 4 is shown in Figure Theore ([8]) For n 1, Γ(R n ) = n. Theore ([8]) For n 4, IR(R n ) = n 4.

27 4.. Bishop 0 (a) γ(r 4 ) = 4 with doinating set in green along the botto row. (b) α(r 4 ) with independent set in green along a ain diagonal. This set is also a independent doinating set. Figure 4.1.1: The doination nuber and independence nuber of an n n Rook graph are equal. Figure 4.1.: The doination nuber of R(4, 5) is Bishop Every paraeter in our doination chain is known for B(n, n) = B n, as seen in Fricke et al. s survey [8]. Bishop graphs are popularly analyzed as Rook graphs rotated 45 degrees because Rook graphs are sipler to exaine as noted in Section 4.1. As seen in Figures 4..1c and 4..1d we can take the sallest whole n n Rook graph that is ebedded for each rotated graph to find the lower bound for the doination nuber of a Bishop graph. Using results fro Cockayne, Gable, and Shepherd [5] and Yaglo and Yaglo [], we have that the lower three paraeters in the doination chain, irredundance nuber, doination nuber, and independent doination nuber, are all equal. Theore ([8]) For n 31, ir(b n ) = n.

28 4.. Bishop 1 Figure 4.1.3: The upper irredundence nuber of R 5 is (5) 4 = 6 with irredundant set in green. Theore 4... ([], [5]) For n 1, γ(b n ) = n. Corollary ([], [5]) For n 1, i(b n ) = n. Yaglo and Yaglo showed that the chosen vertices that coprise an axiu independent set are forced to be picked fro those with saller degree. This can be seen in Figure 4... Theore ([], [18]) All of the bishops in an independent set of axiu size on B n are on the outer ring of squares. While the the independence nuber and upper doination nuber are equal for the Bishop graph on an n n board, the upper irredundance nuber can be shown to be uch larger and follows an interesting pattern. Theore ([], [8]) For n 1, α(b n ) = Γ(B n ) = n. The following result is due to Fricke et al. [8], though a inor correction is needed. In order to satisfy that Γ(G) IR(G) in the doination chain, we ust have n 6. An exaple on a 6 6 board can be seen in Figure Theore For n 6, IR(B n ) = 4n 14

29 4.3. King (a) A 5 5 Bishop graph B 5. (b) B 5 redrawn as a Rook graph rotated 45 degrees. (c) B 5 black coponent rotated 45 degrees. Note the 3 3 Rook graph inside indicating that the doination nuber of the black coponent is at least 3. (d) B 5 white coponent rotated 45 degrees. Note the Rook graph inside indicating that the doination nuber of the white coponent is at least. Figure 4..1: B 5 rotated and divided into its white and black coponents. The edges corresponding to oves between black squares are given in purple, oves between white squares are given in orange. Since the black and white graphs for separate coponents of B 5, we have γ(b 5 ) King Since the oveent of a king forces that the piece can doinate at ost 3 3 section of the board at any one point in tie, it is easier to deterine those paraeters involved in the doination chain. Specifically, for any n n chessboard, we know the doinating nuber, independent doinating nuber, and independence nuber for the associated n n King graph, K(n, n) = K n. Doinating and independent sets for the 5 5 King graph can be seen in Figure

30 4.3. King 3 Figure 4..: α(b 5 ) = 8 with independent set in green. (a) i(b 5 ) = 5 with independent doinating set in green on the iddle row. (b) Γ(B 5 ) = 8 with upper doinating set in green along the border. Figure 4..3: B 5 independent doinating and upper doinating sets. Theore ([], [8]) γ(k n ) = n+. 3 Corollary ([], [8]) i(k n ) = n+. 3 Theore ([], [8]) For n 1, α(k n ) = n+1. Kings doination and independence is easily extended to n boards due to the finite nature of the king oves. Adjusting Theore and Corollary 4.3. for and n yields γ(k(, n)) = K(i(, n)) = + n Adjusting Theore siilarly yields α(k(, n)) = +1 n+1. Unlike with rooks and bishops, the irredundance nubers for the King graph are not fully known. However, all known bounds for both the lower and upper irredundance nubers are due to Favoron et al. [7] as stated below. Figure 4.3. provides an

31 4.3. King 4 Figure 4..4: B 6 with upper irredundance in green. (a) K 5 with independent doinating set. (b) K 5 with independent set. Figure 4.3.1: K 5 with independent doinating set and independent set. exaple of an irredundant set and Figure 4.3.1b gives an exaple of an upper irredundant set. Currently, there are no known results for the upper doination nuber for a king. Theore ([7]) ir(k n ) n+ 3 1 when n 4 od 6. Theore ([7]) n 9 ir(k n ) n+ 3 and so ir(kn ) = n 9 Theore ([7] For n 6, (n 1) 3 IR(K n ) n 3. when n 0 od 3.

32 4.4. Queen 5 Figure 4.3.: ir(k 4 ) = 3 as per Theore with irredundant set in three of the center vertices. 4.4 Queen One of the ost well-known probles, a packing proble, first posed for an 8 8 by chess puzzle coposer Max Bezzel [8], is this: How any queens can be placed on a chessboard so that no queen attacks another? For an n n board the answer is n queens. Thus, one paraeter in our doination chain, the independence nuber, is known for Q(n, n) = Q n. For the other five paraeters, however, only bounds can be given, and uch of what is known is due to coputer searches. Theore ([1]) For n > 3, α(q n ) = n. As in Corollary 4.1.3, we can generalize our queens independence to an n board. We achieve for n > 3, α(q(, n)) = in{, n}. Perhaps because the Queen graph probles are particularly challenging, it is well studied. Such questions that have been raised about this class of graphs include What is the fewest nuber of queens needed to attack or occupy every square on the board? (If we took out the words or occupy we would be looking for the total doination value for a given board). The answer to this posed question for an 8 8 board, is exactly 5 queens, or γ(q 8 ) = 5. However, there is no known generalization of this 5-queen result; this proble reains open for an n n board since a forula has yet to be deterined for the nuber of queens needed. Nevertheless, soe reasonable lower bounds for queens doination on a square board do exist.

33 4.4. Queen 6 Theore ([18]) γ(q n ) 1 (n 1). Corollary ([18]) For n = 4k + 1, γ(q n ) 1 (n + 1) = k + 1. (a) Q 5 with axiu independent set. (b) Q 5 with iniu independent doinating set. Figure 4.4.1: Q 5 independent set and independent doinating set. As stated in Chapter, the queen has been studied using other pieces fro different variations of chess. These new pieces include the dragon king (Dk n ) and dragon horse (Dh n ) pieces fro the Japanese chess gae Shogi [3]. The dragon king and dragon horse cobine the oves of the king with the rook and bishop respectively and, thus, the oveent is closer to that of the queen than any other piece in classic chess. The goal is to use these different chess pieces to find bounds for both the doination nuber and independent doination nuber of their graphs and then use these results to gain greater insight into bounding these sae paraeters for the Queen graph. The oves for these pieces are in Figure 4.4. along with the oveent for the Queen. Theore ([3]) For n 7, γ(dk n ) = i(dk n ) = n 3. Theore ([3]) For n 4, γ(dh n ) n 1. Conjecture ([8]) For n sufficiently large, γ(q n ) = i(q n ). Theore ([19]) For n 5, Γ(Q n ) n 5.

34 4.5. Knight 7 Z0Z0Z 0Z0Z0 Z0s0Z 0Z0Z0 Z0Z0Z Z0Z0Z 0Z0Z0 Z0l0Z 0Z0Z0 Z0Z0Z Z0Z0Z 0Z0Z0 Z0a0Z 0Z0Z0 Z0Z0Z (a) Dragon King (b) Classic Queen (c) Dragon Horse Figure 4.4.: The dragon king has the cobined oves of a rook and king, while the dragon horse has the cobined oves of a bishop and king. These are ost closely related to the queen s oveent, as shown in (b). The original oves of the rook and bishop are in given in blue with the additional king oves in green. n ir(q n ) IR(Q n ) Table 4.1: Irredundance values for n n Queen graph. There are perhaps no good upper bounds for ir(q n ) or IR(Q n ). Several paraeters have been coputer calculated for various n n boards as given in Table 4.1 [1]. 4.5 Knight Since the Knight graph N(n, n) = N n is bipartite, α(n n ) = Γ(N n ) = IR(N n ) as proved by Cockayne et al. [4]. Recall fro Chapter that each successive knight ove forces the knight to change to a square of opposing color. Thus, we can intuitively see that, by placing a knight on every square of one particular color, we can axiize our knight placeent. If n is even, we can choose either of the two colors for this arrangeent, while if n is odd we would choose that color with a larger nuber of squares.

35 4.5. Knight 8 Theore ([4]) α(n n ) = Γ(N n ) = IR(N n ) = n for n even n +1 for n odd We also know the axiization paraeters for the n Knight graph by using n instead of n and extending the result above: α(n(, n)) = Γ(N(, n)) = IR(N(, n)) = n for n even n+1 for n odd In 1987 Hare and Hedetniei [11] published a linear-tie algorith to find the doination nuber for a given Knight graph. This algorith returns the doination nuber for n boards. (a) α(n 4 ) = 8 with independent set along the border in green. (b) γ(n 4 ) = 4 with doinating set on the center vertices in green. Figure 4.5.1: N 4 independent set and doinating set. Interestingly, the knight s n n irredundance nuber is studied only for n = 1,. Moreover, the first three paraeters in the doination chain ir(n), γ(n), and i(n) don t have any good upper bounds. ir(n n ) is virtually unstudied, while i(n n ) values are known for specific n as seen in Table 4. [1].

36 4.5. Knight 9 n ir(n n ) 1 4 γ(n n ) i(n n ) Table 4.: Miniization paraeter values for n n Knight graph.

37 30 Chapter 5 New Results In this Chapter we explore our new doination chain results for the n Bishop graph. Without loss of generality we will assue n since the n Bishop graph is isoorphic to a Bishop graph with n rows and coluns. Definition A n chessboard has rows and n coluns of alternating black and white squares. The botto left square is black. 5.1 Bishop Moveent on an n Graph Consider the n chessboard. We ay denote each square as an ordered pair of whole nubers (x, y), 0 x n 1, 0 y 1. An n board has rows and n coluns. Figure provides an exaple. Proposition Let (x, y) be any square on the n chessboard such that x and y are whole nubers, 0 x n 1, 0 y 1. For an integer k, a bishop placed at (x, y) ay ove to any square of the for: (x + k, y + k) with ax{ y, x} k in{ y 1, n x 1}, k 0 (5.1.1)

38 5.1. Bishop Moveent on an n Graph 31 (0, 3) (0, ) (0, 1) (0, 0) (1, 0) (, 0) (3, 0) (4, 0) Figure 5.1.1: A 4 5 siplified Bishop graph with four rows and five coluns. or (x + k, y k) with ax{ + y + 1, x} k in{y, n x 1}, k 0 (5.1.) Proof. Fro Section.3, a bishop can ove to any square on the sae two diagonals, the northeast to southwest diagonal, odeled by Equation (5.1.1), and the northwest to southeast, Equation (5.1.). Due to this oveent, we just need to show that the bounds given in these equations both contain the bishop on the board and contact an edge, thus showing that the oveent described encopasses all possible squares on the two diagonals. Let (x, y) be a square on the n chessboard, 0 x n 1, 0 y 1. Let us consider the northeast to southwest diagonal. Case 1: Consider Equation (5.1.1). We have ax{ x, y} k in{ y 1, n x 1}. Consider the following subcases: Case 1a: Let x y. Then ax{ y, x} = x and our bound becoes x k in{ y 1, n x 1}, k 0. (Note: The lower bound becoes x k, regardless of whether y 1 or n x 1 is the axiu.) Using this lower bound we obtain (x x, y x), which is (0, y x). Since x 0

39 5.1. Bishop Moveent on an n Graph 3 and x y, (0, y x) is clearly on the board and touches the left edge. If y 1 n x 1, then x k y 1 and the bishop can ove to any square on the diagonal fro (x x, y x) to (x + y 1, y + y 1). That is, fro (0, y x) to (x + y 1, 1). Since x 0, y 1 0, and y 1 n x 1, we obtain 0 x+ y 1 n 1. Thus (x+ y 1, 1) is on the board and touches the top edge. Therefore for x k y 1, Equation (5.1.1) holds. If n x 1 y 1, then x k n x 1 and the bishop can ove to any square on the diagonal fro (x x, y x) = (0, y x)to (x+n x 1, y +n x 1) = (n 1, y + n x 1). Now y 0, so n x 1 0, and n x 1 y 1, and we obtain 0 y + n x 1 n 1. Thus, (n 1, y + n x 1) is on the board and touches the right edge. Therefore for x k n x 1 and Equation (5.1.1) holds. Case 1b: If x > y, then ax{ y, x} = y and the bound under consideration becoes y k in{ y 1, n x 1}, k 0 (Note: The lower bound becoes y k, regardless of whether y 1 or n x 1 is the axiu.) Preceding above we achieve (x y, y y) = (x y, 0). Since y 0 and y x, (x y, 0) is clearly on the board and touches the botto edge. If y 1 n x 1, then y k y 1 and the bishop can ove to any square on the diagonal fro (x y, y y) to (x + y 1, y + y 1), fro (x y, 0) to (x + y 1, 1) As before, (x + y 1, 1) is a square on the top of the board. Therefore for y k y 1, Equation (5.1.1) holds. If n x 1 y 1, then y k n x 1 and the bishop can ove to any square on the diagonal fro (x y, y y) = (x y, 0) to (x+n x 1, y +n x 1) = (n 1, y + n x 1), this latter point which is located on the right edge of the board. Therefore for y k n x 1 and Equation (5.1.1) holds. Case : In considering Equation (5.1.) we have ax{ + y + 1, x} k in{y, n x 1}.

40 5.1. Bishop Moveent on an n Graph 33 Consider the following subcases: Case a: If x y 1, then ax{ +y +1, x} = x and the bound becoes x k in{y, n x 1}, k 0. (Note: The lower bound becoes x k, regardless of whether + y + 1 or n x 1 is the axiu.) With this lower bound, we arrive at (x x, y + x) = (0, y + x). Since x 0, y 0, and x y 1, we get 0 x + y y + y 1 = 1. Then (0, y + x) is on the left edge of the board. If y n x 1, then x k y and the bishop can ove to any square on the diagonal fro (x x, y + x) to (x + y, y y), so fro (0, y + x) to (x + y, 0). Now since, x+y 0 and y n x 1, we get 0 x+y x+n x 1 = n 1. Thus (x + y, 0) is on the board and touches the botto edge. Therefore for x k y, Equation (5.1.) holds. If n x 1 y, then x k n x 1 and the bishop can ove to any square on the diagonal fro (x x, y + x) to (x + n x 1, y n + x + 1), that is fro (0, y + x) to (n 1, y n + x + 1). As before, (0, y +x) is on the board and touches the left edge. Since y n x 1, y (n x 1) = y n + x + 1 0, and since 1 y n x 1 we get y (n x 1) = y n+x+1 1. Then (n 1, y n+x+1) is on the board and touches the right edge. Therefore for x k n x 1, Equation (5.1.) holds. Case b: Let y 1 x. We have ax{ +y+1, x} = +y+1 and the equality becoes +y+1 k in{y, n x 1}, k 0. (Note: The lower bound becoes + y + 1 k, regardless of whether y 1 or n x 1 is the axiu.) Using the lower bound where k = +y+1, we achieve (x +y+1, y+ y 1), or (x + y + 1, 1). Since x y 1, we get x + y and since x n 1 we get x ( y 1) = x +y +1 n 1. Thus, (x +y +1, 1) is on the board and touches the top edge of the board.

41 5.. Bishop Graph Definitions 34 If y n x 1, then + y + 1 k y and the bishop can ove to any square on the diagonal fro (x + y + 1, y + y 1) to (x + y, y y), that is fro (x + y + 1, 1) to (x + y, 0). In case a (x + y, 0) was shown to be on the botto of the board. Therefore for + y + 1 k y, Equation (5.1.) holds. If n x 1 y, then + y + 1 k n x 1 and the bishop can ove to any square on the diagonal fro (x +y+1, y+ y 1) to (x+n x 1, y n+x+1), that is fro (x +y+1, 1) to (n 1, y n+x+1). Fro case a, (n 1, y n+x+1) is on the right edge of the board. Therefore for + y + 1 k n x 1, Equation (5.1.) holds. 5. Bishop Graph Definitions Definition The Bishop graph for an n chessboard, B(, n), has vertex set V (B(, n)) = {(x, y) 0 x n 1, 0 y 1} with edge set E(B(, n)) containing all edges of the for (x, y) (w, z) where vertex (x, y) can ove to vertex (w, z) as prescribed in Proposition Note: Unless otherwise stated, assue 0 x n 1 and 0 y 1 Definition 5... For a vertex (x, y) of B(, n), a positive diagonal containing (x, y) also contains the set of of vertices that satisfy Equation (5.1.1). Definition For a vertex (x, y) of B(, n), a negative diagonal containing (x, y) also contains the set of of vertices that satisfy Equation (5.1.). Definition The positive diagonal graph is the subgraph B + (, n) with vertex set V (B(, n)) and edge set consisting of those edges between any two vertices on the sae positive diagonal.

42 5.. Bishop Graph Definitions 35 Definition The negative diagonal graph is the subgraph B (, n) with vertex set V (B(, n)) and edge set consisting of those edges between any two vertices on the sae negative diagonal. Note: Every vertex lays on exactly one positive diagonal and one negative diagonal. Definition A graph is positively covered by vertex set S if S contains at least one vertex fro each positive diagonal in the graph. Definition A graph is negatively covered by vertex set S if S contains at least one vertex fro each negative diagonal in the graph. Definition The origin (0, 0) is the lower left square on the Bishop graph. The origin is a black square. Definition The Black Bishop graph B b (, n) is the coponent of the Bishop graph containing the origin. That is, B b (, n) is the induced subgraph of B(, n) such that (x, y) V (B b (, n)) if and only if x + y is even. Definition The black positive diagonal graph is the induced subgraph B + b (, n) of B + (, n) where v V (B + b (, n)) if and only if v V (B b(, n)). Definition The black negative diagonal graph is the induced subgraph B b (, n) of B (, n) where v V (B b (, n)) if and only if v V (B b(, n)). We observe that B b (, n) represents the range of otion on the black squares of an n chessboard. B + b (, n) and B b (, n) represent positively directed oveent and negatively directed oveent on the black squares, respectively. Definition The White Bishop graph, B w (, n), is the coponent of the Bishop graph not containing the origin. That is, B w (, n) is the induced subgraph of B(, n) such that (x, y) V (B w (, n)) if and only if x + y is odd.

43 5.3. Independence of the n Bishop Graph 36 Definition The white positive diagonal graph is the induced subgraph B + w (, n) of B + (, n) where v V (B + w (, n)) if and only if v V (B w (, n)). Definition The white negative diagonal graph is the subgraph B w (, n) of B (, n) where v V (B w (, n)) if and only if v V (B w (, n)). B w (, n) represents the range of otion on the white squares of an n chessboard. B + w (, n) and B w (, n) represent positively directed oveent and negatively directed oveent on the white squares, respectively. 5.3 Independence of the n Bishop Graph The purpose of this section is to establish a forula for the axiu nuber of bishops that can be placed on an n board without attacking, that is, to deterine the packing proble for n bishops. Recall the definition of independence nuber and independent set. We prove a forula for the independence nuber of a Bishop graph, n, and give a construction to find an associated independent set. Before we begin, it is iportant to count the nuber of positive and negative diagonals on the Bishop graph. Proposition B(, n) has + n 1 positive diagonals and + n 1 negative diagonals. Proof. Each positive (negative) diagonal begins fro the left colun or botto (top) row. Because there are n coluns and rows and one corner vertex counted twice, we get n + 1 positive (negative) diagonals. Observe the positive diagonal graph B + (3, 4) and negative diagonal graph B (3, 4) in Figure

44 5.3. Independence of the n Bishop Graph 37 (a) B + (3, 4) (b) B (3, 4) Figure 5.3.1: B + (3, 4) and B (3, 4) each have = 6 diagonals. We will need to break our arguent into three cases, all based on the parity of and n: odd, even with subcases n odd and n even. First let us observe the differences in the black and white coponents of each of these types of graphs in Figure (a) B b (, 3) (b) B b (3, 3) (c) B b (, 4) (d) B w (, 3) (e) B w (3, 3) (f) B w (, 4) Figure 5.3.: Black coponent graphs (top) with white coponent graphs (botto). In Figures 5.3.a and 5.3.d, we can see that the black and white coponent graphs B b (, 3) and B w (, 3) both have two positive and two negative diagonals. On the 3 3 graphs in Figures 5.3.b and 5.3.e, we see that while the two coponent graphs are different, each coponent has the sae nuber of positive and negative diagonals. That is B b (3, 3) has three positive and three negative diagonals and B w (3, 3) has two positive and two negative diagonals. In contrast, in Figure 5.3.c,