61 problems

Let $L$ be a Latin square of order $n$. A transversal is a set of $n$ cells containing exactly one cell from each row, column, and symbol. Ryser's conjecture. Every Latin square of…

Brualdi–Stein conjecture. Every Latin square contains a near transversal.

Let $L$ be a Latin square of order $n$, and let a partial transversal be a set of cells containing no two cells in the same row, column, or with the same symbol. Brualdi's conjectu…

Brualdi–Ryser–Stein conjecture. Every equi-$n$-square contains a transversal of size $n-1$.

A latin hypercube of order $n$ and dimension $d$ is a $d$-dimensional array filled with $n$ symbols so that every line contains all symbols exactly once. A transversal is a diagona…

For each $n\in\mathbb{N}$, let $\mathcal{L}(n)$ be the set of Latin squares of order $n$ using the symbols $[n]=\{1,2,\ldots,n\}$, and let $L_n\in\mathcal{L}(n)$ be chosen uniforml…

Let $0<c<1$ be a constant. For a graph $G$, let $\alpha(G)$ be the size of a largest independent set, and let $h(G)$ be the size of a smallest set meeting every maximum independent…

A Latin array of order $n$ is an $n\times n$ square filled with an arbitrary number of symbols such that no symbol appears twice in the same row or column. A full transversal is a…

Let $n$ be odd, and let the order-$n$ Latin square be the Cayley table of the cyclic group $\mathbb{Z}_n$. Let $T_n$ denote its number of transversals. Vardi's conjecture. For all…

Rodney's conjecture. Every Latin square contains a duplex.

A Latin square of order $n$ is an $n\times n$ array in which each symbol occurs exactly once in every row and column. A partial transversal is a set of cells with no two in the sam…

Kotlar–Ziv conjecture. Every $n \times n$ matroidal Latin square has an independent partial transversal of size $n-1$.

Let an $n\times n$ Latin square be an arrangement of $n$ symbols in $n$ rows and $n$ columns, with each row and column containing each symbol exactly once. A partial transversal is…

Brualdi–Ryser conjecture. Every latin square $L$ of order $n$ possesses a near transversal, and if $n$ is odd then $L$ possesses a transversal.

Let $Q$ be a latin square of order $n$. A partial transversal is a set of entries with no two in the same row, column, or containing the same symbol. Stein–Brualdi conjecture. Ever…

Drisko's conjecture. Either $A$ has a transversal, or $A$ can be transformed into $R_{2n-2,n}$ by permuting rows, columns, and symbols.

n-row transversal conjecture. If $m\ge n^2$, then there exist $n$ rows of $A$ that together are the union of $n$ transversals.

Disjoint-transversal conjecture. If $m\ge2n-1$, then $A$ has $m-(n-1)$ pairwise disjoint transversals.

Let $n\geqslant 28$ be an even integer. A latin square of order $n$ is an $n\times n$ array in which each symbol occurs exactly once in each row and column, and a transversal is a…

A latin square of order $n$ is an $n\times n$ array in which each of $n$ symbols occurs exactly once in each row and column. A transversal is a selection of $n$ entries containing…

Let $0<c<1$ be a constant. For a graph $G$, let $\operatorname{MIS}(G)$ denote its family of maximal independent sets, and define the family of large maximal independent sets by ……

Beta conjecture. For all graphs $G$ on $n$ vertices,

Odd-order transversal conjecture. Every Latin square of odd order possesses a transversal.

Let $M(d,n)$ denote the set of Latin hypercubes of dimension $d$ and order $n$. For all even $d$ and even $n>2$, let $H\in M(d,n)$ be a hypercube constructed by replacing the order…

Let $M(d,n)$ denote the set of Latin hypercubes of dimension $d$ and order $n$. A transversal is a set of $n$ entries containing each value in every coordinate position and each sy…