45 problems

Let $L$ be a Latin square of order $n$. A partial transversal is a set of cells containing no two cells in the same row, column, or with the same symbol, and a full transversal has…

A latin square of order $n$ is an $n\times n$ array filled with $n$ symbols so that each row and column contains every symbol exactly once. A transversal is a set of $n$ entries co…

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

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…

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…

Brualdi–Ryser–Stein conjecture. Every $n\times n$ Latin square has a partial transversal of size $n-1$.

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

Stein's conjecture. Let $R$ be an $(n-1)\times n$ Latin array. Then $R$ contains a transversal.

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

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

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

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…

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…

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,

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…

Hiss–O'Brien conjecture. If $H$ admits a $G$-invariant transversal, then

An equi-$n$-square is an $n\times n$ array filled with $n$ symbols, each appearing exactly $n$ times. A transversal is a collection of cells sharing no row, column, or symbol; its…

A latin $n$-cuboid of order $q$ is an $n$-dimensional array with one dimension of length less than $q$ and the remaining dimensions of length $q$, whose lines contain different sym…

Zeon formulation of Ryser's conjecture I. One has

Commutator-intersection conjecture. If $H$ is abelian and $(G,H,T)$ is an RCC loop folder, then

Let $L$ be a Latin square of order $n$. For $1\leqslant k\leqslant n$, let $E_k$ be the number of diagonals of $L$ containing exactly $k$ distinct symbols. Let $t_{ij}$ be the numb…

Let $L$ be a Latin square of even order $n$. Let $w$, $x$, $y$, and $z$ be the numbers of transversals in $L$ of types $T^{000}$, $T^{011}$, $T^{101}$, and $T^{110}$, respectively.…

Let $G$ be a finite non-Abelian group, and let $L(G)$ denote its latin square. A partial transversal is a collection of cells that intersects each row, column, and symbol class at…

Rodney's conjecture. Every Latin square has a 2-transversal.