Completing some Partial Latin Squares

AbstractWe show that any partial 3 r× 3 r Latin square whose filled cells lie in two disjoint r×r sub-squares can be completed. We do this by proving the more general result that any partial 3 r by 3 r Latin square, with filled cells in the top left 2r× 2 r square, for which there is a pairing of the columns so that in each row there is a filled cell in at most one of each matched pair of columns, can be completed if and only if there is some way to fill the cells of the top left 2 r× 2 r square

Completing partial latin squares with prescribed diagonals

AbstractThis paper deals with completion of partial latin squares L=(lij) of order n with k cyclically generated diagonals (li+t,j+t=lij+t if lij is not empty; with calculations modulo n). There is special emphasis on cyclic completion. Here, we present results for k=2,…,7 and odd n⩽21, and we describe the computational method used (hill climbing). Noncyclic completion is investigated in the cases k=2,3 or 4 and n⩽21

Completing partial latin squares with one nonempty row, column, and symbol

Let r,c,s ∈{1,2,…,n} and let PP be a partial latin square of order n in which each nonempty cell lies in row r, column c, or contains symbol s. We show that if n ∉ {3, 4, 5} and row r, column c, and symbol s can be completed in P, then a completion of P exists. As a consequence, this proves a conjecture made by Casselgren and Häggkvist. Furthermore, we show exactly when row r, column c, and symbol s can be completed

Completing partial Latin squares with two filled rows and two filled columns

It is shown that any partial Latin square of order at least six which consists of two filled rows and two filled columns can be completed

Some applications of matching theorems

PhDThis thesis contains the results of two investigations. The rst concerns the 1- factorizability of regular graphs of high degree. Chetwynd and Hilton proved in 1989 that all regular graphs of order 2n and degree 2n where > 1 2 ( p 7 1) 0:82288 are 1-factorizable. We show that all regular graphs of order 2n and degree 2n where is greater than the second largest root of 4x6 28x5 71x4 + 54x3 + 88x2 62x + 3 ( 0:81112) are 1-factorizable. It is hoped that in the future our techniques will yield further improvements to this bound. In addition our study of barriers in graphs of high minimum degree may have independent applications. The second investigation concerns partial latin squares that satisfy Hall's Condition. The problem of completing a partial latin square can be viewed as a listcolouring problem in a natural way. Hall's Condition is a necessary condition for such a problem to have a solution. We show that for certain classes of partial latin square, Hall's Condition is both necessary and su cient, generalizing theorems of Hilton and Johnson, and Bobga and Johnson. It is well-known that the problem of deciding whether a partial latin square is completable is NP-complete. We show that the problem of deciding whether a partial latin square that is promised to satisfy Hall's Condition is completable is NP-hard