Latin Squares Of Odd Order Research Articles

On the upper embedding of Steiner triple systems and Latin squares

It is proved that for any prescribed orientation of the triples of either a Steiner triple system or a Latin square of odd order, there exists an embedding in an orientable surface with the triples forming triangular faces and one extra large face.

A note on rainbow matchings in strongly edge-colored graphs

The Ryser Conjecture which states that there is a transversal of size n in a Latin square of odd order n is equivalent to finding a rainbow matching of size n in a properly edge-colored Kn,n using n colors when n is odd. Let δ be the minimum degree of a graph. Wang proposed a more general question to find a function f(δ) such that every properly edge-colored graph of order f(δ) contains a rainbow matching of size δ, which currently has the best bound of f(δ)≤3.5δ+2 by Lo. Babu, Chandran and Vaidyanathan investigated Wang’s question under a stronger color condition. A strongly edge-colored graph is a properly edge-colored graph in which every monochromatic subgraph is an induced matching. Wang, Yan and Yu proved that every strongly edge-colored graph of order at least 2δ+2 has a rainbow matching of size δ. In this note, we extend this result to graphs of order at least 2δ+1.

Balanced diagonals in frequency squares

We say that a diagonal in an array is λ-balanced if each entry occurs λ times. Let L be a frequency square of type F(n;λ); that is, an n×n array in which each entry from {1,2,…,m=n∕λ} occurs λ times per row and λ times per column. We show that if m⩽3, L contains a λ-balanced diagonal, with only one exception up to equivalence when m=2. We give partial results for m⩾4 and suggest a generalization of Ryser’s conjecture, that every Latin square of odd order has a transversal. Our method relies on first identifying a small substructure with the frequency square that facilitates the task of locating a balanced diagonal in the entire array.

MAXIMUM GENUS EMBEDDINGS OF LATIN SQUARES

AbstractIt is proved that every non-trivial Latin square has an upper embedding in a non-orientable surface and every Latin square of odd order has an upper embedding in an orientable surface. In the latter case, detailed results about the possible automorphisms and their actions are also obtained.

ON THE CHROMATIC INDEX OF LATIN SQUARES

A proper coloring of a Latin square of order n is an assignment of colors to its elements triples such that each row, column and symbol is assigned n distinct colors. Equivalently, a proper coloring of a Latin square is a partition into partial transversals. The chromatic index of a Latin square is the least number of colors needed for a proper coloring. We study the chromatic index of the cyclic Latin square which arises from the addition table for the integers modulo n . We obtain the best possible bounds except for the case when n/ 2 is odd and divisible by 3. We make some conjectures about the chromatic index, suggesting a generalization of Ryser’s conjecture (that every Latin square of odd order contains a transversal).

Existence of rainbow matchings in strongly edge-colored graphs

The famous Ryser Conjecture states that there is a transversal of size n in a latin square of odd order n, which is equivalent to finding a rainbow matching of size n in a properly edge-colored Kn,n when n is odd. Let δ denote the minimum degree of a graph. In 2011, Wang proposed a more general question to find a function f(δ) (f(δ)≥2δ+1) such that for each properly edge-colored graph of order f(δ), there exists a rainbow matching of size δ. The best bound so far is f(δ)≤3.5δ+2 due to Lo. Babu et al. considered this problem in strongly edge-colored graphs in which each path of length 3 is rainbow. They proved that if G is a strongly edge-colored graph of order at least 2⌊3δ4⌋, then G has a rainbow matching of size ⌊3δ4⌋. They proposed an interesting question: Is there a constant c greater than 34 such that every strongly edge-colored graph G has a rainbow matching of size at least cδ if |V(G)|≥2⌊cδ⌋? Clearly, c≤1. We prove that if G is a strongly edge-colored graph with minimum degree δ and order at least 2δ+2, then G has a rainbow matching of size δ.

New complete Latin squares of odd order

Let m be an odd powerful number. We show that there is a sequenceable group of order 3m and hence a complete Latin square of that order. Further, the sequencings we construct are starter-translate and so they may be combined with themselves and other sequencings to construct complete Latin squares for many more orders. We also consider square-free m with all prime divisors congruent to 1 modulo 6: there is a sequenceable group, and hence a complete Latin square, of order 3m in this case too. This work gives 104 new orders less than 10,000 for which a complete Latin square is known to exist, the smallest of which is 75.

Parity Types, Cycle Structures and Autotopisms of Latin Squares

The parity type of a Latin square is defined in terms of the numbers of even and odd rows and columns. It is related to an Alon-Tarsi-like conjecture that applies to Latin squares of odd order. Parity types are used to derive upper bounds for the size of autotopy groups. A new algorithm for finding the autotopy group of a Latin square, based on the cycle decomposition of its rows, is presented, and upper bounds for the size of autotopy groups are derived from it.

TRANSVERSALS, INDIVISIBLE PLEXES AND PARTITIONS OF LATIN SQUARES

A k-plex in a latin square of order n is a selection of kn entries in which each row, column and symbol occurs exactly k times. A 1-plex is also called a transversal. A k-plex is indivisible if it contains no c-plex for 0 1. For n ∈ {2, 6}, the existence of a latin square of order n with a partition into 1- plexes was shown by Bose, Shrikhande and Parker [1]. A natural extension of the result to k-plexes is the following. We prove that if k is a proper divisor of n, then there exists a latin square of order n composed of parallel indivisible k-plexes. Define κ(n) to be the largest integer k such that some latin square of order n contains an indivisible k-plex. A conjecture by Rodney [3, p143] says that every latin square contains a 2-plex, which implies that κ(n) 2. We show that for all n > 2, there exists a latin square of order n with two parallel indivisible ⌊ n 2 ⌋-plexes. A corollary is that κ(n) ≥ ⌊ n 2 ⌋ for all n > 2. We report on extensive computations of the indivisible partitions of the latin squares of order n ≤ 9. Up to order 8 we count the number of indivisible partitions of every type. Due to Bose, Shrikhande and Parker [1], and Finney [10] for n = 6, we know that there exists a latin square for each order n > 2 which possesses a k-plex for all 0 ≤ k ≤ n. Wanless [12] showed that there are latin squares of every even order which contain no odd plexes, where an odd plex is a k-plex such that k is odd. It has been conjectured by Rodney [4], and by Wanless [12], that every latin square has a set of ⌊ n 2 ⌋ parallel 2-plexes which implies that every latin square of odd order has a k-plex for each 0 ≤ k ≤ n. We show that among the latin squares of even order there are many other possibilities concerning the existence of odd k-plexes. 1 We prove that for all even n > 2, there exists a latin square of order n which has no k-plex for any odd k 3, there exists a latin square of order n with at least one entry not in any transversal. Such a latin square is called a confirmed bachelor and illustrates a restriction on the transversals in a latin square. In our study of transversals we consider bachelor latin squares in greater detail and other types of restrictions that might occur. We give a concise alternative proof of the result by Wanless and Webb. A main result is that for all even n ≥ 10, except perhaps if n is a power of 2, there exists a latin square of order n that possesses a transversal, but every transversal coincides on a single entry. A theorem by Wanless, which was prompted by our data, shows that there exist arbitrarily large latin squares of odd order in which the proportion of entries not in a transversal is asymptotic to one ninth. We report on computations of parallel transversals in the latin squares of order 9. Thus we prove that the above mentioned conjectures by Rodney and Wanless are true for order 9, and that every latin square of order 9 has at least 3 parallel transversals. Our computations suggest that the constructions which prove our theorems illustrate rare behaviour in large latin squares. The computations also give further evidence in support of a conjecture by Ryser [11] (see [3, p143]) that every latin square of odd order has a transversal. Many of the theorems in this thesis rely on a simple but powerful lemma [6, 9]. The lemma gives a necessary condition for the existence of k-plexes. To show the existence of k-plexes we use constructive techniques. It remains open to establish a sufficient condition for the existence of k-plexes in latin squares. The main results of this thesis can be found in [2, 5–8]. Keywords : latin square, transversal, plex, duplex, triplex, quadruplex, indivisible plex, indivisible partition, orthogonal partition, bachelor square, mutually orthogo- nal latin squares, partial latin square, protoplex, homogeneous partial latin square, latin trade, homogeneous latin trade, rainbow factor, graph factorisation, quasi- group, complete mapping, orthomorphism.

On the range of influences in back-circulant Latin squares

In this note we obtain a large lower bound for the index of a certain critical set in the back-circulant Latin squares of odd order. This resolves in the negative a conjecture of Fitina, Seberry and Chaudhry [Back-circulant Latin square and the influence of a set, Austral. J. Combin. 20 (1999) 163–180].

Subsquare-free Latin squares of odd order

For every odd positive integer m we prove the existence of a Latin square of order 3m having no proper Latin subsquares. Combining this with previously known results it follows that subsquare-free Latin squares exist for all odd orders.

An application of group theory to the existence and nonexistence of orthogonal latin squares

SUMMARY Three theorems on the construction of orthogonal latin squares are presented. The first describes a general procedure for constructing orthogonal latin squares of odd order. The second theorem proves that the procedure of theorem one fails for some families of latin squares of even order. The last theorem demonstrates that certain families of latin squares of even order are orthogonally mateless.