A Generalization of Plexes of Latin Squares

A $k$-plex of a latin square is a collection of cells representing each row, column, and symbol precisely $k$ times. The classic case of $k=1$ is more commonly known as a transversal. We introduce the concept of a $k$-weight, an integral weight function on the cells of a latin square whose row, column, and symbol sums are all $k$. We then show that several non-existence results about $k$-plexes can been seen as more general facts about $k$-weights and that the weight-analogues of several well-known existence conjectures for plexes actually hold for $k$-weights

Among graphs, groups, and latin squares

A latin square of order n is an n × n array in which each row and each column contains each of the numbers {1, 2, . . . , n}. A k-plex in a latin square is a collection of entries which intersects each row and column k times and contains k copies of each symbol. This thesis studies the existence of k-plexes and approximations of k-plexes in latin squares, paying particular attention to latin squares which correspond to multiplication tables of groups. The most commonly studied class of k-plex is the 1-plex, better known as a transversal. Although many latin squares do not have transversals, Brualdi conjectured that every latin square has a near transversal—i.e. a collection of entries with distinct symbols which in- tersects all but one row and all but one column. Our first main result confirms Brualdi’s conjecture in the special case of group-based latin squares. Then, using a well-known equivalence between edge-colorings of complete bipartite graphs and latin squares, we introduce Hamilton 2-plexes. We conjecture that every latin square of order n ≥ 5 has a Hamilton 2-plex and provide a range of evidence for this conjecture. In particular, we confirm our conjecture computationally for n ≤ 8 and show that a suitable analogue of Hamilton 2-plexes always occur in n × n arrays with no symbol appearing more than n/√96 times. To study Hamilton 2-plexes in group-based latin squares, we generalize the notion of harmonious groups to what we call H2-harmonious groups. Our second main result classifies all H2-harmonious abelian groups. The last part of the thesis formalizes an idea which first appeared in a paper of Cameron and Wanless: a (k,l)-plex is a collection of entries which intersects each row and column k times and contains at most l copies of each symbol. We demonstrate the existence of (k, 4k)-plexes in all latin squares and (k, k + 1)-plexes in sufficiently large latin squares. We also find analogues of these theorems for Hamilton 2-plexes, including our third main result: every sufficiently large latin square has a Hamilton (2,3)-plex

Equitable partitions of Latin-square graphs

Funding: R.A. Bailey and Peter J. Cameron are grateful to Shanghai Jiao Tong University for funding, from the National Science Foundation of China (11671258) and STCSM (17690740800), a research visit where part of this study was done. Alexander L. Gavrilyuk was supported by BK21plus Center for Math Research and Education at Pusan National University, Republic of Korea. Sergey V. Goryainov was supported by the National Science Foundation of China, STCSM (17690740800) and RFBR (17‐51‐560008).We study equitable partitions of Latin-square graphs and give a complete classification of those whose quotient matrix does not have an eigenvalue -3.PostprintPeer reviewe

Approximate Transversals of Latin Squares

A latin square of order n is an n by n array whose entries are drawn from an n-set of symbols such that each symbol appears precisely once in each row and column. A transversal of a latin square is a subset of cells that meets each row, column, and symbol precisely once. There are many open and difficult questions about the existence and prevalence of transversals. We undertake a systematic study of collections of cells that exhibit regularity properties similar to those of transversals and prove numerous theorems about their existence and structure. We hope that our results and methods will suggest new strategies for the study of transversals. The main topics we investigate are partial and weak transversals, weak orthogonal mates, integral weight functions on the cells of a latin square, applications of Alon\u27s Combinatorial Nullstellensatz to latin squares, and complete mappings of finite loops

Nonextendible Latin Cuboids

We show that for all integers m >= 4 there exists a 2m x 2m x m latin cuboid that cannot be completed to a 2mx2mx2m latin cube. We also show that for all even m > 2 there exists a (2m-1) x (2m-1) x (m-1) latin cuboid that cannot be extended to any (2m-1) x (2m-1) x m latin cuboid

Latin Squares and Their Applications to Cryptography

A latin square of order-n is an n x n array over a set of n symbols such that every symbol appears exactly once in each row and exactly once in each column. Latin squares encode features of algebraic structures. When an algebraic structure passes certain latin square tests , it is a candidate for use in the construction of cryptographic systems. A transversal of a latin square is a list of n distinct symbols, one from each row and each column. The question regarding the existence of transversals in latin squares that encode the Cayley tables of finite groups is far from being resolved and is an area of active investigation. It is known that counting the pairs of permutations over a Galois field ��pd whose point-wise sum is also a permutation is equivalent to counting the transversals of a latin square that encodes the addition group of ��pd. We survey some recent results and conjectures pertaining to latin squares and transversals. We create software tools that generate latin squares and count their transversals. We confirm previous results that cyclic latin squares of prime order-p possess the maximum transversal counts for 3 ≤ p ≤ 9. Furthermore, we create a new algorithm that uses these prime order-p cyclic latin squares as building blocks to construct super-symmetric latin squares of prime power order-pd with d \u3e 0; using this algorithm we accurately predict that super-symmetric latin squares of order-pd possess the confirmed maximum transversal counts for 3 ≤ pd ≤ 9 and the estimated lower bound on the maximum transversal counts for 9 \u3c pd ≤ 17. Also, we give some conjectures regarding the number of transversals in a super-symmetric latin square. Lastly, we use the super-symmetric latin square for the additive group of the Galois field (��32, +) to create a simplified version of Grøstl, an iterated hash function, where the compression function is built from two fixed, large, distinct permutations