An analogue of Ryser's Theorem for partial Sudoku squares

In 1956 Ryser gave a necessary and sufficient condition for a partial latin rectangle to be completable to a latin square. In 1990 Hilton and Johnson showed that Ryser's condition could be reformulated in terms of Hall's Condition for partial latin squares. Thus Ryser's Theorem can be interpreted as saying that any partial latin rectangle $R$ can be completed if and only if $R$ satisfies Hall's Condition for partial latin squares. We define Hall's Condition for partial Sudoku squares and show that Hall's Condition for partial Sudoku squares gives a criterion for the completion of partial Sudoku rectangles that is both necessary and sufficient. In the particular case where $n=pq$, $p|r$, $q|s$, the result is especially simple, as we show that any $r \times s$ partial $(p,q)$-Sudoku rectangle can be completed (no further condition being necessary).Comment: 19 pages, 10 figure

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

An Evans-style result for block designs

For positive integers $n$ and $k$ with $n \geq k$, an $(n,k,1)$-design is a pair $(V, \mathcal{B})$ where $V$ is a set of $n$ points and $\mathcal{B}$ is a collection of $k$-subsets of $V$ called blocks such that each pair of points occur together in exactly one block. If we weaken this condition to demand only that each pair of points occur together in at most one block, then the resulting object is a partial $(n,k,1)$-design. A completion of a partial $(n,k,1)$-design $(V,\mathcal{A})$ is a (complete) $(n,k,1)$-design $(V,\mathcal{B})$ such that $\mathcal{A} \subseteq \mathcal{B}$. Here, for all sufficiently large $n$, we determine exactly the minimum number of blocks in an uncompletable partial $(n,k,1)$-design. This result is reminiscent of Evans' now-proved conjecture on completions of partial latin squares. We also prove some related results concerning edge decompositions of almost complete graphs into copies of $K_k$.Comment: 17 pages, 0 figure

New 2--critical sets in the abelian 2--group

In this paper we determine a class of critical sets in the abelian {2--group} that may be obtained from a greedy algorithm. These new critical sets are all 2--critical (each entry intersects an intercalate, a trade of size 4) and completes in a top down manner.Comment: 25 page

Completion and deficiency problems

Given a partial Steiner triple system (STS) of order $n$, what is the order of the smallest complete STS it can be embedded into? The study of this question goes back more than 40 years. In this paper we answer it for relatively sparse STSs, showing that given a partial STS of order $n$ with at most $r \le \varepsilon n^2$ triples, it can always be embedded into a complete STS of order $n+O(\sqrt{r})$, which is asymptotically optimal. We also obtain similar results for completions of Latin squares and other designs. This suggests a new, natural class of questions, called deficiency problems. Given a global spanning property $\mathcal{P}$ and a graph $G$, we define the deficiency of the graph $G$ with respect to the property $\mathcal{P}$ to be the smallest positive integer $t$ such that the join $G\ast K_t$ has property $\mathcal{P}$. To illustrate this concept we consider deficiency versions of some well-studied properties, such as having a $K_k$-decomposition, Hamiltonicity, having a triangle-factor and having a perfect matching in hypergraphs. The main goal of this paper is to propose a systematic study of these problems; thus several future research directions are also given