The Pennsylvania State University The Graduate School ORTHOGONALITY AND EXTENDABILITY OF LATIN SQUARES AND RELATED STRUCTURES

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1 The Pennsylvania State University The Graduate School ORTHOGONALITY AND EXTENDABILITY OF LATIN SQUARES AND RELATED STRUCTURES A Dissertation in Mathematics by Serge C. Ballif c 2012 Serge C. Ballif Submitted in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy May 2012

2 The dissertation of Serge C. Ballif was reviewed and approved by the following: Gary L. Mullen Professor of Mathematics Dissertation Advisor Chair of Committee James Sellers Professor of Mathematics W. Dale Brownawell Distinguished Professor of Mathematics James L. Rosenberger Professor of Statistics John Roe Professor of Mathematics Department Head Signatures are on file in the Graduate School.

3 Abstract Two of the most important topics in the study of latin squares are questions of orthogonality and extendability. A latin square of order n is an n n array consisting of n distinct symbols such that each of n symbols occurs precisely once in each row and column. Two latin squares are said to be orthogonal if no two cells contain the same ordered pair of symbols when the squares are superimposed. There are many generalizations of latin squares, and in these generalizations there is a natural notion of orthogonality. In particular, we can view a latin square as a coloring of a graph. We say that two colorings of a graph are orthogonal if, whenever two vertices share a color in one coloring, then they have a different color in the other coloring. It is well known that there cannot be more than n 1 pairwise orthogonal latin squares of order n. Given a graph, G, we seek a bound on the maximum size of a set of pairwise orthogonal colorings of G. We derive several upper bounds based on parameters of the graph such as the number of vertices and edges, the maximum degree of a vertex, or the existence of large cliques. As a consequence we establish upper bounds on the maximum cardinality of a set of pairwise orthogonal colorings for several latin structures including latin rectangles, row latin squares, single diagonal latin squares, and double diagonal latin squares. We show that these bounds are the best possible. Questions about the extendability of latin squares are related to obtaining a latin square from a partially filled latin square. A partial latin square of order n is an n n array consisting of n symbols such that each of n symbols occurs at most once in each row and column. It is an NP-complete problem to determine whether a partial latin square can be completed to a latin square of the same order. In 1974 Alan Cruse derived necessary and sufficient conditions to extend a partial latin rectangle to a latin square. Here we provide an alternate proof of Cruse s Theorem. Then we use the tools of this new and different proof to prove an analogous theorem for iii

4 frequency squares. A frequency square of type F(n; λ 1,..., λ k ) is an n n array filled with k symbols if the symbol i occurs in each row and column precisely λ i times. A partial transversal of size r in a latin square is a set of r cells representing distinct rows, columns, and symbols. We show that the questions of orthogonality and extendability overlap with the question of the existence of a (partial) transversal of latin squares. The problem of finding a (partial) transversal in a latin square can be recast in terms of extending portions of a latin square or in terms of the parameters of the graph obtained from a coloring that defines the latin square. We extend the well known conjectures of Bualdi and Ryser on the existence of (partial) transversals and propose a few tools for tackling these conjectures. iv

5 Table of Contents List of Tables List of Symbols List of Definitions Acknowledgments vi vii viii x Chapter 1 Introduction Definition of a Latin Square Completing Partial Latin Squares Orthogonal Latin Squares Constructions of MOLS MOLS via Bivariate Polynomials The Kronecker Product Chapter 2 Orthogonal Colorings of Graphs Generalizing Latin Squares Latin Squares As Graph Colorings Orthogonal Colorings of Graphs An Upper Bound for N(G, n) Based on Vertex Degree An Upper Bound for N(G, n) Based on Cliques An Improved Upper Bound for N(G, n) Based on Vertices of High Degree An Upper Bound for N(G, n) Based on the Average Vertex Degree An Upper Bound for N(G, n) Based on the Number of Edges v

6 2.9 Constructions with Orthogonal Colorings Coloring-Extended Supergraphs MOLS via Mutually Orthogonal Colorings Or-Products Chapter 3 Latin Polyominoes Coloring Polyominoes Mutually Orthogonal Latin Polyominoes Mutually Orthogonal Latin Rectangles Based on Extra Symbols Mutually Orthogonal Latin Rectangles Orthogonal Rectangles With Extra Symbols Kronecker Products and Mutually Orthogonal Latin Rectangles of Type (r, s, t) Applications of Mutually Orthogonal Colorings The Social Golfer Problem The (r, s, t)-social Golfer Problem Kirkman School Girl Problem Chapter 4 Embedding Partial Latin Structures Completing Partial Latin Rectangles Generalizations of Cruse s Theorem Completing Frequency Rectangles Saturated Rectangles of Type (r, s, t) Proof of Cruse s Theorem The Shuffle Lemma Proof of Theorem 48; a Frequency Square Version of Cruse s Theorem Proof of Theorem 52; a Saturated Rectangle Version of Cruse s Theorem 54 Chapter 5 Transversals in Latin Structures Quasiembeddings of Latin Squares Cruse s Theorem for Partial Latin Squares Quasi-embeddings and Brualdi s Conjecture Transverals and Extremal Graph Theory Chapter 6 Further Work Summary vi

7 6.2 Further Work Appendix A König s Theorem For Polyominos 64 A.1 König s Theorem Appendix B Upper Bounds on N(G, n) for Common Graphs 67 Bibliography 71 vii

8 List of Tables 1.1 Upper bounds for the cardinality of mutually orthogonal sets of latin structures B.1 Upper bounds for N(G, n) viii

9 List of Symbols N(n) N P (n) The maximum possible cardinality of a set of a set of mutually orthogonal latin squares of order n, p. 4 The maximum possible cardinality of a set of mutually orthogonal latin squares of order n via bivariate polynomials modulo n p. 7 G The complement of G, p. 12 N(G, n) The maximum possible cardinality of a set of mutually orthogonal n- colorings of a graph G, p. 15 G H The cartesian product of graphs G and H, p. 13 L(G) The line graph of G, p. 14 Oχ(G) The orthogonal chromatic number of G, p. 15 G C1,...,C k A coloring-extended supergraph obtained from mutually orthogonal colorings C 1,..., C k of G, p. 26 G H The or-product of graphs G and H, p. 29 N(m, n) N(r, s, t) The maximum number of mutually orthogonal m n latin rectangles, p. 36 The maximum number of mutually orthogonal latin rectangles of type (r, s, t), p. 37 ix

10 List of Definitions Latin Square p. 1 Partial Latin Square p. 2 Extension of a Partial Latin Square p. 2 Orthogonal Latin Squares p. 3 MOLS p. 4 Transversal p. 5 Kronecker Product p. 8 Polyomino p. 11 Latin Polyomino p. 11 Graph p. 12 Vertex p. 12 Edge p. 12 Adjacent p. 12 Degree p. 12 Complement p. 12 Subgraph p. 16 n-coloring p. 12 Chromatic Number p. 12 x

11 Rook s Graph p. 13 Cartesian Product p. 13 Orthogonal Colorings p. 14 Degree Bound p. 17, p. 21 Equi-n Square p. 17 Row (or Column) Latin Square p. 17 Single Diagonal Latin Square p. 17 Double Diagonal Latin Square p. 17 Gerechte Design p. 19 Sudoku Square p. 19 Average-Degree Bound p. 23 Edge Bound p. 24 Coloring-Extended Supergraph p. 26 Or-Product of Graphs p. 29 Diagonal Polyomino p. 35 Latin Rectangle p. 36 Latin Rectangle of Type (r, s, t) p. 37 Partial Latin Rectangle of Type (r, s, t) p. 46 xi

12 Acknowledgments I am thankful for the patience and encouragement of my advisor, Gary Mullen. He has been a constant support through my graduate studies. He introduced me to the beautiful world of latin squares, and for that I shall be forever grateful. I want to thank my wife, Jenny, and my kids, Andrew, Annabelle, and Eliza. They mean the world to me. I m proud to call them mine. xii

13 Dedication This thesis is dedicated to my sweetheart, Jenny, the love of my life. xiii

14 3 Orthogonal Sudoku Squares Mutually Orthogonal Colorings of the Petersen Graph xiv

15 Chapter1 Introduction 1.1 Definition of a Latin Square A latin square of order n is an n n array consisting of n distinct symbols so that each symbol occurs exactly once in each row and column. For example the square is a latin square of order 4. Latin squares occur naturally in many structures such as group multiplication tables. In fact, latin squares are precisely the multiplication tables of an algebraic structure called a quasigroup. A quasigroup is a set, Q, together with a binary operation : Q Q Q such that for each a, b Q there exist unique values x, y Q that are solutions to the equations a y = b and x a = b. The existence (and uniqueness) of such solutions is equivalent to the quasigroup multiplication table having permutations for each row and each column. Any n symbols may be used in a latin square of order n, and exchanging one symbol for another does not fundamentally change the properties of a latin square. For convenience we shall take the numbers 1, 2,..., n as the symbols unless otherwise stated. Latin squares are beautiful combinatorial structures that have a wide range of applications including experimental design, tournament scheduling, finite geometries, and cryptography. Perhaps the best known example of latin squares are sudoku

16 2 squares. The focus of this thesis is not on the applications of latin squares, but rather on various combinatorial questions about latin squares and related structures. We shall define many latin structures throughout this thesis, and most definitions will be provided throughout the thesis. To make things easier for the reader we have provided a List of Definitions on page viii that provides page numbers where key terms are defined. In this thesis we investigate generalizations of two of the great problems in latin squares, namely completing partial latin squares and finding sets of orthogonal latin squares. In Chapter 2 we extend the notion of orthogonal latin squares to colorings of graphs. We obtain upper bounds on the maximum cardinality of a set of mutually orthogonal colorings of a graph. We show that each of these upper bounds gives us the well known upper bounds on the cardinality of a set of latin squares and on sets of related latin structures. In Chapter 3 we consider mutually orthogonal colorings when the graph is a polyomino. In Chapter 4 we extend Cruse s Theorem on completing a partial latin rectangle to a latin square [12] in two theorems. The first of these theorems gives necessary and sufficient conditions to extend a partial frequency rectangle to a frequency square. The second theorem gives conditions to extend a partial latin rectangle to a partial latin rectangle with the maximum possible number of entries. We give a brief introduction to these topics in the next two sections. In Section 1.4 we provide a few constructions of sets of mutually orthogonal latin squares that will be used later in the thesis. We also extend a result of Rivest [30] on the non-existence of a pair of orthogonal latin squares of order n = 2 w derived from polynomials modulo 2 w. We prove an exact bound on the maximum size of a set of mutually orthogonal latin squares of order n for any n, and show that the result of Rivest is a special case of our bound. 1.2 Completing Partial Latin Squares A partial latin square of order n is an n n array (possibly with empty cells) based on n distinct symbols such that each row and column contains each of the n symbols at most once. Latin squares are special cases of partial latin squares. An extension of a partial latin square P is a (partial) latin square P such that both P and P share

17 3 the same entry in row i and column j whenever that cell is nonempty in P. We say that a partial latin square P is embedded inside a partial latin square P if the entries of P are preserved in the corresponding location of P. Example 1. Below we display embeddings of a partial latin square P inside squares P and P P P P P Both P and P are maximal in the sense that they cannot be embedded (nontrivially) inside another partial latin square of the same order. The question of whether a partial latin square can be completed to a latin square of the same order is known to be NP-complete [10]. Example 1 illustrates how a greedy algorithm (starting from the bottom right) can fail. In Chapter 4 we investigate some necessary and sufficient conditions to extend a partially filled latin rectangle to a latin square. We extend these conditions to more general structures such as frequency squares and latin rectangles. 1.3 Orthogonal Latin Squares Two latin squares of order n are said to be orthogonal if, when superimposed, each of the n 2 possible ordered pairs of symbols occur. Up to a permutation of symbols there are exactly two latin squares of order 3, and , and they are orthogonal to each other. Note that if two latin squares are orthogonal, then permuting the symbols in one of the latin squares still yields an orthogonal pair. Latin squares were first studied by Euler in the now famous 36 officer problem [14]: 36 soldiers of 6 different ranks belong to 6 regiments, and each of the 6 regiments consists of 6 soldiers of different ranks. Is it possible to arrange the officers into a 6 6 square such that no regiment or rank appears more than once in any row or column?

18 4 The 36 officer problem is equivalent to finding a pair of orthogonal latin squares of order 6. The symbols of the first square would correspond to 6 different ranks, and the symbols from the second square would correspond to the 6 different regiments. Euler was unable to find a pair of orthogonal latin squares of order 6. However, Euler did find orthogonal pairs of latin squares of every order except order 4k + 2. The claim that there exists no pair of orthogonal latin squares of order 4k + 2 became known as The Euler Conjecture. There can be no orthogonal pair of latin squares of order 2 because opposite corners of a latin square of order 2 must contain the same symbol. It was not until 1900 that it was proved by Tarry that no orthogonal pair of order 6 latin squares exists (see [14, p. 140]). Tarry ruled out all possible pairs through a nearly exhaustive search. In 1960, the Euler Conjecture was proved false by Bose, Parker, and Shrikhande [6] for all orders except 2 and 6. A set of pairwise orthogonal latin squares is said to be a set of mutually orthogonal latin squares (MOLS). We use the notation N(n) for the maximum size of a set of MOLS of order n. The exact value of N(n) is known only when n = 6 or when n is a power of a prime. However, it is well-known that N(n) n 1 [14, 25]. A set of n 1 MOLS of order n is called a complete set of MOLS. Dozens of papers have been written with the purpose of increasing the lower bound for N(n) for various values of n. The reference [11] is a survey paper that describes several constructions of sets of MOLS. The paper [34] showed that for sufficiently large n, N(n) n 1/17 2. Thus it is known that N(n) as n. In this thesis we view latin squares as a coloring of a graph, and we find upper bounds on the cardinality of a set of pairwise orthogonal colorings of a graph. In Chapter 2 we prove (in multiple ways) that N(n) n 1. We also introduce several combinatorial structures related to latin squares and prove similar bounds on the maximum possible cardinality of orthogonal sets. In Table 1.1 we summarize some of the upper bounds that we obtain. In Appendix B we display a complete summary of the upper bounds obtained in this thesis.

19 5 Graph G Table 1.1: We list the upper bounds for the cardinality of mutually orthogonal sets of latin structures. bounds that are known to be best possible are in bold. Degree Bound (n 2 m d 1)l nl l m Average- Degree Bound (v D 1)n v n r(s j) r+s n Upper Clique Edge Bound Bound ( 2) e v (n r)( n v 2 )+r( n v 2 ) Equi-n Square n + 1 n + 1 n + 1 Row or Column Latin Square of n n n n order n Latin Square of order n n 1 n 1 n 1 n 1 Single Diagonal Latin Square of order n Double Diagonal Latin Square of odd order, n > 3 Sudoku square of order n m n Latin Rectangle m n n 2 n 2 n 2 n 2 n 3 n 2 n 3 n 2 n 2 n 2 n n n 2 n 1 n 1 n 1 A transversal of a latin square is a subset of n cells such that each of the n rows, columns, and symbols is represented in one of the cells. A partial transversal of size r is a subset of r cells such that each of the n rows, columns, and symbols is represented in at most one of the cells. Below we display a transversal in a latin square of order In Chapter 5 we investigate transversals and their relationship to orthogonality and extendability of latin squares and related structures.

20 6 1.4 Constructions of MOLS In this section we introduce a few of the standard constructions of MOLS, namely bivariate polynomials and the Kronecker product. We discuss some limitations of these constructions and extend a theorem of Rivest from [30] MOLS via Bivariate Polynomials A latin square of order n based on symbols from a set S determines a function f : S 2 S on a finite set S of size n > 0 such that for any fixed a S both functions f (a, y) and f (x, a) are permutations of S. Then the two latin squares obtained from the functions f 1 (x, y), f 2 (x, y) are orthogonal if the pairs (f 1 (x, y), f 2 (x, y)) are distinct for each ordered pair (x, y) S S. When n is a power of a prime, one can obtain a complete a set of MOLS of order n using a finite field, F n, of order n. For each a F we define the function n f a (x, y) = ax + y. Then the squares obtained from the functions f a (x, y) form a complete set of MOLS of order n (see [25]). For example we can create 3 MOLS using the finite field of order 4 obtained by adjoining α to /2, where α 2 +α+1 = 0. The substitution 0 1, 1 2, α 3, and α gives the three MOLS L 1 = , L α = , and L α+1 = The only known cases when the bound N(n) = n 1 is attained is when n = q is power of a prime [25, p. 21]. Theorem 2. For q a prime power the set of polynomials f a (x, y) = ax + y with 0 a q represents a complete set of q 1 MOLS of order q. Hence N(q) = q 1. The claim that a complete set of n 1 MOLS of order n exists only when n is a power of a prime is known as the Prime Power Conjecture. The Prime Power Conjecture is easy to understand, yet difficult to prove. In [28] Mullen proposed the Prime Power Conjecture as a candidate for the next Fermat problem to replace Fermat s Last Theorem.

21 7 Now we consider latin squares where the function f is a bivariate polynomial f (x, y) [x, y] and entries are considered modulo n. Let N P (n) denote the maximal number of MOLS given by polynomials modulo n. In [30] it was shown that there is no pair of polynomials f 1 (x, y), f 2 (x, y) modulo 2 w (w 1) that forms a pair of orthogonal latin squares (i.e. N P (2 w ) = 1). Here we generalize this result for latin squares modulo n. Theorem 3. Let n be an integer, and let p be the smallest prime dividing n. Then N P (n) = p 1. Proof. We start by showing that there exist p 1 MOLS modulo n of order n given by polynomials. For each 1 a p 1, define f a (x, y) = ax + y. We claim that the squares represented by the polynomials f a (x, y) are mutually orthogonal. Choose a and b such that 1 a b p 1 and suppose that the same ordered pair occurs in two coordinates (f a (x, y), f b (x, y)) = (f a (x, y ), f b (x, y )). Then by definition (ax + y, bx + y) = (ax + y, bx + y ) so subtracting the equations ax + y = ax + y bx + y = bx + y yields (a b)x (a b)x (mod n) so (a b)(x x ) 0 (mod n). The number (a b) is a zero divisor modulo n if and only if (a b) divides n. However, 0 < a b < p, so (a b) can t divide n. From this we conclude that x x = 0, so x = x. It then follows that y = y. Thus the squares f a (x, y) and f b (x, y) are mutually orthogonal, so we have provided a collection of p 1 MOLS modulo n. Next we show that p 1 is maximal. Let m = n/p. We shall show that each n n latin square f (x, y) (mod n) consists of m 2 latin subsquares of size p p. Moreover, any latin square g(x, y) that is orthogonal to f (x, y) also shares the same m 2 subsquares in such a way that the p p latin squares must be MOLS. Since there are at most p 1 MOLS of order p, the result will follow. Assume first that f (x) is a permutation polynomial modulo n. We first consider the value f (x + km) for k an integer. Using the binomial expansion and collecting

22 8 terms, we see that for each 1 x n, f (x + km) = f (x) + jm for some integer j. Now consider a latin square obtained from a polynomial f (x, y) modulo n. By considering each variable separately, we get the inclusion modulo n f (x + km, y + lm) {f (x, y), f (x, y) + m,..., f (x, y) + (p 1)m} (k, l ). Thus f consists of m 2 latin subsquares of order p. If we fix x and y then the entries of this latin subsquare are f (x, y), f (x, y) + m,...,f (x, y) + (p 1)m (where these entries are reduced modulo n). Any other latin square g(x, y) modulo n also consists of the same latin subsquares of order p. If f and g are polynomial MOLS modulo n of order n, then f and g also provide MOLS when restricted to the p p subsquares. Since there are at most (p 1) MOLS of order p, there can be at most (p 1) MOLS modulo n of order n. Example 4. The square obtained from the function f (x, y) = 4x + 2 y (mod 9) is pictured below. f (x, y) = 4x + 2 y (mod 9) The shaded cells illustrate one of the many latin subsquares. Any other latin square obtained from a polynomial modulo 9 will have a latin subsquare formed by the same cells (though not necessarily using the same symbols) The Kronecker Product The Kronecker product can be used to create a latin square from smaller latin squares. In general the Kronecker product of two matrices A and B of sizes i j

23 9 and k l respectively is the ik jl matrix a 11 B a 1j B A B =..... a i1 B a i j B a 11 b 11 a 11 b 12 a 11 b 1l a 1 j b 11 a 1 j b 12 a 1 j b 1l a 11 b 21 a 11 b 22 a 11 b 2l a 1 j b 21 a 1 j b 22 a 1 j b 2l a 11 b k1 a 11 b k2 a 11 b kl a 1j b k1 a 1 j b 12 a 1j b kl = a i1 b 11 a i1 b 12 a i1 b 1l a i j b 11 a i j b 12 a i j b 1l a i1 b 21 a i1 b 22 a i1 b 2l a i j b 21 a i j b 22 a i j b 2l a i1 b k1 a i1 b k2 a i1 b kl a i j b k1 a i j b 12 a i j b kl It is well known that the Kronecker product of a latin square A = (a i j ) of order m with a latin square B = (b kl ) of order n is a latin square of order mn based on the ordered pairs of symbols (a i j, b kl ). The Kronecker product construction provides a good method for creating MOLS. In general if A and A are orthogonal latin squares, and B and B are orthogonal latin squares, then A B and A B are orthogonal latin squares [25]. Example 5. The pairs of MOLS A = , A = and B = , B =

24 10 can be used to create the MOLS A B = and A B displayed below A B = A B = The Kronecker product construction of MOLS gives us a lower bound on N(n) [25]. Theorem 6. N(mn) min{n(m), N(n)} Proof. If A 1,..., A k are MOLS of order m and B 1,..., B k are MOLS of order n, then A 1 B 1,..., A k B k are MOLS of order mn. Corollary 7. If n = q 1 q k where q 1 < q 2 < < q k are powers of distinct primes, then N(n) q 1 1. The bound of Corollary 7 is an improvement on the bound of Theorem 3. Corollary 7 is sometimes called MacNeish s Theorem [14, p. 390].

25 Chapter2 Orthogonal Colorings of Graphs 2.1 Generalizing Latin Squares The latin property of a latin square forbids any cells in the same row or column from sharing the same symbol. There are many combinatorial structures that have this latin property. Example 8. A polyomino is a (possibly disconnected) subset of cells from a square tiling of the plane. If we label the cells of a polyomino so that the latin propery is satisfied, then we get a latin polyomino. As an example, below we exhibit five latin polyominoes in two polyomino shapes Note that the two latin polyominoes on the left are orthogonal in the sense that when they are superimposed each of the ordered pairs is distinct. Similarly, the three latin polyominoes on the right are mutually orthogonal. In each case we have exceeded the bound of N(2) = 1 that occurs in the case where the polyomino shape is a square. In this chapter we consider latin squares as a special case of a proper n-coloring of a simple graph. We show that several known results about orthogonal latin squares are special cases of orthogonal colorings of graphs, a notion first studied

26 12 in [9]. As a corollary, we obtain bounds on the cardinality of orthogonal sets of several structures including equi-n squares, row or column latin squares, latin squares, single diagonal latin squares, and double diagonal latin squares. In Sections 2.2 and 2.3 we show how a latin square is a coloring of a graph and generalize the notion of orthogonality to colorings of graph. Sections 2.4 through 2.8 contain upper bounds on the size of a set of orthogonal graph colorings based on graph parameters such as the average vertex degree, maximum degree, maximum clique size, and number of edges. In Section 2.9 we introduce some uses of orthogonal graph colorings as well as some graph constructions that aid our understanding of the structure of orthogonal graph colorings. In Section 5.2 we show that the problem of finding a transversal in a latin square is equivalent to the problem of finding a clique in a certain graph. This equivalence of problems leads to a new possible line of attack on proving the conjectures of Brualdi and Ryser on the existence of (partial) transversals in a latin square. 2.2 Latin Squares As Graph Colorings A graph, G, is a set of vertices, V (G), and a set, E(G), of unordered pairs of distinct elements of V (G) called edges. The order of G is the number of vertices in V (G). Two vertices are said to be adjacent (or neighbors) if they are connected by an edge in E. We write u v to signify that u is adjacent to v. The degree of a vertex is the number of vertices adjacent to it. The complement of a graph G is a graph G that has vertex set V (G), and two vertices of G are adjacent if and only if they are not adjacent in G. A (proper) coloring of G is a labeling of the vertices so that any two adjacent edges have distinct labels (which we call colors). An n-coloring is a coloring consisting of (at most) n colors. It is usually convenient to assume these colors are the numbers 1,..., n. A graph, G, is said to be n-colorable if there exists an n-coloring of G. The chromatic number χ = χ(g), of G is the minimum number, n, such that there exists an n-coloring of G. If C is a coloring of G, then we write C(v) for the color that C assigns to v. For example, below we display a 3-coloring of a famous graph called the Petersen

27 13 Graph. It can be shown that the chromatic number of the Petersen graph is 3 (i.e., there exists a 3-coloring but no 2-coloring). A latin square of order n can be viewed as a proper n-coloring of an n n rook s graph. An m n rook s graph is a graph that shows the legal moves that a rook may make in a game of chess on an m n board. The squares are the vertices, and two squares are adjacent if they share a row or column. An m n rook graph is a special case of the cartesian product of complete graphs K m K n. A complete graph, K n, is a graph of order n with all vertices adjacent. The Cartesian product of graphs G and H is the graph G H that has vertex set V (G) V (H), and vertices (g, h) and (g, h ) are adjacent in G H if one of the following two conditions holds: g = g and h h in H, h = h and g g in G. As an example, the cartesian product = of two graphs of order 3 is a graph of order 9. As the number of edges increases, the cartesian product is harder to visualize. A rook s graph such as K 3 K 5 K 3 K 5 = is complicated to draw or visualize, but we can take advantage of its uniform struc-

28 14 ture by picturing it as an 3 5 grid where we consider two cells to be adjacent if and only if they share a row or a column. Another construction of a rook s graph squares requires us to define two graph theory terms: the complete bipartite graph, K m,n, and the line graph, L(K m,n ). A bipartite graph is a graph whose vertices can be divided into two disjoint sets U and V such that every edge connects a vertex in U to one in V. The complete bipartite graph, K m,n, is a bipartite graph where every vertex of a set of size m is connected to every vertex of a set of size n. A line graph, L(G) of a graph, G, is the graph that uses the edge set of G as its set of vertices. Two vertices of L(G) are adjacent if and only if their corresponding edges in E(G) share a common vertex in G. We obtain an m n rook s graph as the line graph of the complete bipartite graph, K m,n. Below we show the graph K 3,5 K 3,5 L(K 3,5 ) 3 5 grid One nice feature about the line graph construction of a rook s graph is that any subgraph of K m,n has a polyomino as its line graph. The converse also holds. In particular, the line graph has a cell in row i and column j if and only if the i-th vertex in the m-set of K m,n is adjacent to the j-th vertex of the n-set in K m,n. 2.3 Orthogonal Colorings of Graphs We saw in the previous section that latin squares are a special case of the more general concept of the coloring of a graph. Next we show that the notion of orthogonality in latin squares is also a special case of orthogonality in the colorings of a graph. Two colorings, C 1 and C 2, of a graph, G, are said to be orthogonal if whenever u and v are distinct vertices of G, we have distinct ordered pairs of colors (C 1 (u), C 2 (u)) (C 1 (v), C 2 (v)). A set of pairwise orthogonal n-colorings of G is called a set of mutually orthogonal colorings. We denote the maximum size of a

29 15 set of mutually orthogonal n-colorings of a graph G by N(G, n). Example 9. Below we display orthogonal colorings of a graph of order 9 and a graph of order 16. Much of this thesis is devoted to finding an upper bound for N(G, n). Appendix B lists many of the upper bounds that are obtained throughout this thesis. A graph with a pair of orthogonal n-colorings can have at most n 2 vertices because there are at most n 2 distinct ordered pairs of labels. In fact, if any color shows up more than n times in a coloring, then the coloring has no orthogonal mate. At the other extreme we may have an overabundance of colors if n V (G) then each vertex can be assigned a different color. Such a coloring is orthogonal to any coloring, including itself. In the reference, [9], Caro and Yuster investigate the number of colors needed for a graph to have orthogonal colorings. The authors define the orthogonal chromatic number of G, Oχ(G), to be the minimum number of colors in any pair of orthogonal colorings of G. Similarly the k-orthogonal chromatic number of G is the number, Oχ k (G), of colors required so that there exist k mutually orthogonal colorings of G. The authors find bounds on the values of Oχ(G) and Oχ k (G) in terms of the parameters of G such as the maximum degree of any vertex of G or the chromatic number of G. They also show that several classes of graphs always have the lowest possible value Oχ(G) = V (G). Here we are concerned with the function N(G, n) which is related to the function Oχ k (G) by the inequalities N(G, Oχ k (G)) k and Oχ N(G,n) (G) n. Many authors have studied edge colorings of graphs as opposed to vertex colorings. A proper edge coloring of a graph is a labeling of the edges of a graph such that any two edges that share a vertex receive distinct labels. Each edge coloring of

30 16 G corresponds to a vertex coloring of L(G) in a natural way, so any question about edge coloring can be converted into a question about vertex coloring. The converse statement does not hold because most graphs are not the line graph of any graph. Despite the fact that each edge coloring may be viewed as an example of vertex coloring, many questions about edge colorings are interesting in their own right and have been studied in the context of edge colorings. Of particular note to us is the article [3] by Archdeacon and Dinitz where they studied orthogonal edge colorings of graphs. In their article they viewed a latin square as a special case of an edge coloring of the complete bipartite graph K n,n. Results about orthogonal edge colorings can be converted to results about orthogonal vertex coloring by studying the corresponding line graph. Remark 10. A subgraph, H, of a graph, G, is a graph such that V (H) V (G) and E(H) E(G). It will always be the case that if H is a subgraph of G, then N(H, n) N(G, n) because mutually orthogonal colorings of G are also mutually orthogonal colorings of H. 2.4 An Upper Bound for N(G, n) Based on Vertex Degree In this section we introduce a theorem that gives up upper bound on N(G, n) by using a vertex of high degree. We shall show that when G is an n n rook s graph we obtain precisely the bound N(n) n 1. Theorem 11. Let G be a graph of order n 2 with a vertex of degree d. Then n 2 1 d N(G, n). n 1 Proof. Suppose that C 1,..., C k are mutually orthogonal n-colorings of G and that vertex u has neighbors u 1,..., u d. After relabeling we can assume that C 1 (u) = C 2 (u) = = C k (u). Then no vertex in the set G \ {u, u 1,..., u d } can have color C 1 (u) in more than one of the colorings C 1,..., C k, yet at least n 1 vertices in V (G) \ {u, u 1,..., u d } are colored C 1 (u) in each of C 1,..., C m. Thus k(n 1) n 2 1 d so k n2 1 d n 1.

31 17 We shall call the bound of Theorem 11 the degree bound. The degree bound can be applied to several variations of latin squares, some of which we now define. An equi-n square is an n n array such that each of n symbols occurs in exactly n cells. A row (resp. column) latin square of order n is an n n array where each of n symbols occurs in each row (resp. column). A single diagonal latin square is a latin square where each symbol on the main diagonal is distinct. A double diagonal latin square is a single diagonal latin square where each symbol along the back diagonal is also distinct. The degree bound can be applied to an n n rook s graph to obtain the bound N(n) n 1. The degree bound can also be applied to obtain bounds on the size of orthogonal sets of each of the latin structures defined above. Moreover, for each of these structures the degree bound provides the best possible upper bound. Corollary 12. Let n 2 be a positive integer. (1) No set of mutually orthogonal equi-n squares of order n consists of more than n + 1 squares. (2) No set of mutually orthogonal row (or column) latin squares of order n consists of more than n squares. (3) No set of mutually orthogonal latin squares consists of more than n 1 squares. (4) No set of mutually orthogonal single diagonal latin squares consists of more than n 2 squares. (5) No set of mutually orthogonal double diagonal latin squares of odd order n > 3 consists of more than n 3 squares. (See [18]) Proof. In each of these structures, the cells are the vertices, and two cells are adjacent if the structure forbids that the cells share a color. In an equi-n square, no cells are adjacent, so the maximum degree of any vertex in an equi-n square is 0. For statement (1) the degree bound is n + 1. In a row latin square the rows form an n-clique, and the maximum degree of any vertex is n 1. Thus for (2) the degree bound is n2 1 (n 1) = n. n 1 In a latin square each vertex is adjacent to all others in the same row and column, so each vertex has degree 2n 2. Thus for (3) the degree bound is n2 1 (2n 2) n 1 = n 1.

32 18 In a single diagonal latin square, the upper left cell is adjacent to each vertex in the top row, the left column, or the main diagonal, so it has degree 3n 3. In statement (4) we have a degree bound of n2 1 (3n 3) = n 2. n 1 In a double diagonal latin square of odd order the center cell is adjacent to 4n 4 cells, so we have a degree bound of n2 1 (4n 4) n 1 = n 3 for (5). 2.5 An Upper Bound for N(G, n) Based on Cliques An r-clique is a (sub)graph of r pairwise adjacent vertices. Theorem 13. Let r, s, and n be integers satisfying 1 < r, s n < r + s. Let G be a graph with an s-clique, A, that is disjoint from an r-clique, B, such that each vertex in B is adjacent to at least j vertices in A. Then r(s j) N(G, n). r + s n Proof. Let C 1,..., C k be mutually orthogonal n-colorings of G. We may assume without loss of generality that the vertices of A are colored with the same colors, say 1,..., s, so that each pair of colors (1, 1), (2, 2),..., (s, s) occurs in A for each pair of superimposed colorings C i, C j. Among the colorings C 1,..., C k, no vertex of B can have any color in the set {1,..., s} more than once, so colors from the set {1,..., s} can be applied to a vertex of B at most s j times across all the colorings C 1,..., C k. Thus, colors from the set {s + 1,..., n} will occur at least k (s j) times for each of the r vertices of B among the colorings C 1,..., C k. The total number of times that colors from the set {s + 1,..., n} occur in B among the colorings C 1,..., C k is at least r(k s + j). On the other hand, at most n s of the colors from the set {s+1,..., n} can occur in B for each coloring C i. Thus the total number occurances of a color from the set {s + 1,..., n} in B among the colorings C 1,..., C k is at most k(n s). Therefore r(k s + j) m(n s) so that k r(s j) r + s n.

33 19 We shall call the bound of Theorem 13 the clique bound. The clique bound is useful when G has a large clique. Example 14. Below we display two orthogonal 4-colorings of a graph, G. G has a 1-clique that is adjacent to 2 vertices of a 4-clique, so using r = 1, s = 4, and j = 2 the clique bound is N(G, 4) (1)(4 2) = 2. Alternatively G has a 2-clique where each vertex is adjacent to 2 vertices of a 3-clique, so using r = 2, s = 3, and j = 2 we obtain the same clique bound N(G, 4) 2(3 2) = 2. Example 15. There will often be many possible choices for the values of r, s, and j. In the graph, G, of order 16 from Example 9 we can take r = 3, s = 4, and j = 2. We have identified such a 3-clique and such a 4-clique. In this case the clique bound is N(G, 4) 2. The assumptions of the clique bound seem very specific, but the clique bound can be applied to a wide range of structures. The first we mention is a gerechte design. A gerechte design of order n is an n n array that has been partitioned into n regions of size n and filled with n symbols so that each row, column, and region contains each symbol exactly once. A sudoku square of order n 2 is a gerechte design of order n 2 where the partitioned regions are the n 2 subarrays (starting from the top left corner) of size n n. As a corollary to Theorem 13 we get a bound on the size of a set of mutually orthogonal gerechte designs. The following bound was proved in [4, Corollary 2.2].

34 20 Corollary 16. Given a partition of an n n array into regions S 1,..., S n, each of size n, the size of a set of mutually orthogonal gerechte designs for this partition is at most n j, where j is the maximum size of the intersection of a row or column with one of the sets S 1,..., S n with j < n. Proof. Let A be a row (or column). If S i intersects A in j cells (j < n), then we let B = A \ S i. Then with r = B, and s = t = n the clique bound limits the number of mutually orthogonal gerechte designs of order n with partition S 1,..., S n to B (n j) B +n n = n j. Example 17. Consider the graphs G and H below, G : H : where both G and H are partitioned into 5 sets. By Corollary 16 a set of pairwise orthogonal gerechte designs of shape G can consist of at most 2 colorings, while H does not admit an orthogonal pair of colorings. In this case, further investigation will show that G actually does not admit an orthogonal pair of colorings. Corollary 18. The maximum size of a set of mutually orthogonal sudoku squares of order n 2 is n 2 n. Proof. A sudoku square is a gerechte design. The upper left block intersects the top row in n cells, so the clique bound on the cardinality of a set of mutually orthogonal n 2 -colorings is precisely n 2 n. When n is a power of a prime the upper bound of Corollary 18 is always obtainable (see [4] or [29] for constructions). We also note that most of the results of Corollary 12 follow as a corollary to Theorem 13. Alternative Proof of items (2) (5) of Corollary 12. If n is odd let v be the cell in the center of the square, and if n is even let v be the bottom right cell. Let A be the top row of cells, and let B = {v}. We apply Theorem 13 with r = 1 and s = n to get an upper bound of n j, where j is the number of cells in the top row that are adjacent to v. For a row latin square j = 0. For a latin square j = 1. For a single diagonal latin square j = 2, and for a double diagonal latin square of odd order j = 3.

35 An Improved Upper Bound for N(G, n) Based on Vertices of High Degree In fact we can strengthen the degree bound when there are several vertices of high degree and the order of the graph is close to n 2. Theorem 19. Let G be a graph of order n 2 m that has l vertices of degree at least d where 0 m l n. Then (n 2 m d 1)l N(G, n). nl l m Proof. Let C 1,..., C k be mutually orthogonal n-colorings of G, and let each of the vertices u 1,..., u l have degree at least d. Since there are n 2 m total vertices, then in any coloring, C i, there must be at least n m colors applied to n vertices, while the other m colors are applied to an average of n 1 vertices. The number of vertices of one of the colors C i (u 1 ),..., C i (u l ) is at least m(n 1) + (l m)n = ln m. Thus at least one of the vertices u 1,..., u l must share a color with nl m l average across the colorings C 1,..., C k. Therefore, Solving for k yields nl m 1 k n 2 m d 1. l k (n2 m d 1)l. nl l m 1 vertices on Note that when m = 0 we have the degree bound from Theorem 11. For this reason we shall also call the bound of Theorem 19 the degree bound. Example 20. The graph of order 15 = below has a 2-clique of vertices, each with degree 6. The degree bound of Theorem 19 limits the number of mutually orthogonal 4-

36 colorings to ( )(2) (2)(4) = 3.2 = 3. In contrast, the clique bound of Theorem 13 gives a bound of 6 (using a 3-clique and a 2-clique), while the original degree bound of Theorem 11 does not apply. 2.7 An Upper Bound for N(G, n) Based on the Average Vertex Degree The degree bound depends upon the existence of a clique where the degrees of the clique members are high. The following theorem takes into account the average degree of the vertices of a graph. It also provides the best possible bounds for the size of mutually orthogonal latin structures given in Corollary 12. Theorem 21. Let G be a graph of order v with average vertex degree D. Then for n < v, (v D 1)n N(G, n). v n Proof. Let C 1,..., C k be mutually orthogonal n-colorings of G. If we knew that each vertex shared a color with s other vertices in each coloring (on average), then we could argue that sk + D v 1 because we can t eliminate every possible pair of vertices from sharing a color because n < v. The theorem will be proved by finding a lower bound for s. Consider a fixed coloring, C i. We expect that color c will be applied to v/n vertices on average. Therefore, if the color c is selected at random, then we expect each vertex of color c to share a color with v 1 vertices on average. In other words, n if we let l j be the number of vertices of color j (so l l n = v), then the average value of l 1 1, l 2 1,..., l n 1 is v n 1. Next we claim that s v 1. The actual average number, s, of vertices sharing a n color with a given vertex is a weighted average of the values l 1 1, l 2 1,..., l n 1. Specifically, the l i vertices with color i each share a color with l i 1 vertices, so s = l 1(l 1 1) + l 2 (l 2 1) + + l n (l n 1). v In this case we have weighted the larger values l j 1 with greater weights and the smaller values of l j 1 with lesser weights, so s v n 1. Thus v n 1 k + D v 1

37 23 so (v D 1)n k. v n We call the bound of Theorem 21 the average-degree bound. Example 22. Below we display three mutually orthogonal 4-colorings of a graph, G. The average degree of each vertex is 4, so the average-degree bound is N(G, 4) (8 4 1)(4) 8 4 = 3. In contrast, the clique bound is N(G, 4) 4 (using r = 3, s = 3, j = 0 or r = 2, s = 3, j = 1) so the average-degree bound outperforms the clique bound in this instance. The degree bound does not apply because the order of G is less than = 12. Corollary 23. The bounds established in parts (1), (2), (3), and (4) of Corollary 12 hold. Proof. In an equi-n square the average degree is 0, so the average-degree bound is N(G, n) (n2 0 1)n n 2 n = n + 1. In a row latin square the average degree is n 1, so the average-degree bound is N(G, n) (n2 (n 1) 1)n n 2 n = n. In a latin square the average degree is 2(n 1), so the average-degree bound is N(G, n) (n2 2(n 1) 1)n n 2 n In a single diagonal latin square the average degree is 2(n 1) + n 1 n 2 n 2. = n 1., so N(G, n) 2.8 An Upper Bound for N(G, n) Based on the Number of Edges The average-degree bound has the desirable property that it applies to any graph of order between n + 1 and n 2. In fact we can improve the average-degree bound by considering the total number of edges.

38 24 Theorem 24. Let G be a graph with v vertices and e edges, v > n. Write v = qn + r with 0 r < n. Then N(G, n) v 2 e (n r) v n 2 + r v n 2. Proof. Let C 1,..., C k be mutually orthogonal n-colorings of G. Each pair of vertices that share a color in C i must have distinct colors in C j, j i. Thus once each pair of vertices has shared a color in one of the C i s it is not possible to find an additional n-coloring orthogonal to each of the C i s. Thus we need to identify the minimum number of pairs of vertices that will share a color in a given C i. If a color is applied to m vertices, then that accounts for m 2 pairs. Therefore, if color t is applied to µt vertices, then µ1 µ2 µn (2.1) pairs of vertices are accounted for in the coloring C i. The condition µ 1 + +µ n = v guarantees that the expression (2.1) is minimized when n r of the colors appear v times and the other r colors appear v times. Therefore n n Solving for k then yields the theorem. v n k (n r) v + r n v + e We call the bound of Theorem 24 the edge bound. Example 25. The edge bound is always an improvement on the average-degree bound. For example we exhibit 5 mutually orthogonal 4-colorings of a graph below. The average-degree bound for N(G, 4) is 8, while the edge bound for N(G, 4) is 5. Corollary 26. The bounds established in parts (1), (2), (3), and (4) of Corollary 12 hold.

39 25 Proof. In each case we just need to identify the number of edges in the underlying graph G determined by which cells are forbidden to share a color. For an equi-n square G has 0 edges. For a row (or column) latin square G has n n 2 edges. For a latin square G has 2n n n 2 edges. For a single diagonal latin square G has (2n + 1) 2 edges. In each of these cases we can plug these values in for e in the edge bound to obtain parts (1), (2), (3), and (4) of Corollary 12. We have derived several upper bounds on the size of a set of mutually orthogonal n-colorings of a graph. Moreover, we have showed that each of these bound can be used to obtain the bound N(n) n 1 for latin squares that inspired the generalization to graph colorings. Each bound has some advantages and disadvantages. The reader is encouraged to look at Appendix B for a comparison of the various upper bounds for N(G, n). Here we summarize the strengths and weaknesses of each bound. Two of the bounds make use of a large number of edges locally. The degree bound of Theorem 19 makes use of a vertex (or vertices) of high degree, but it applies only to graphs of order close to n 2. The clique bound of Theorem 13 applies to graphs of various sizes, but it depends upon the existence of a large clique (or cliques). Two of the bounds make use of the number of edges globally. The averagedegree bound of Theorem 21 and the edge bound of Theorem 24 are particularly nice because they can be applied to any graph of order between n + 1 and n 2. For many graphs these bounds agree. For instance the polyomino on the left in Example 8 has degree bound, clique bound, average-degree bound, and edge bound equal to 2, while the polyomino on the right in Example 8 has degree bound, average-degree bound, and edge bound equal to 3, but the clique bound is undefined. The bounds on the various latin structures in Corollary 12 have appeared in many places in the literature. However, as far as the author is aware each of the proofs of the results of Corollary 12 that have appeared in print have used an argument based on the existence of an n-clique (and is a variation on the reasoning of the clique bound). Example 27. The bounds of this paper are not always best possible. For example