Latin squares with extreme transversal intersection structure: pairwise-intersecting transversals with no pinned entry, and dominant transversals

Source: Ghafari and Wanless, arXiv:2607.17547 (2026). For a latin square of order $n$, a transversal is a selection of $n$ entries meeting each row, column, and symbol exactly once; an entry is pinned if it lies in every transversal, and a transversal is dominant if it intersects every other transversal. The paper proves that for $n=28$ and every even $32\le n\le 10000$ there is a latin square with at least two transversals, no two of which are disjoint, yet no entry is pinned, and conjectures this for all even $n\ge 28$ (its Conjecture 3); it also proves that squares with a dominant transversal exist for every $n>4$ with $n\not\equiv 3\pmod 4$ and $n\ne 6$, and asks (its Question 1) whether they exist for every $n\ge 7$. The source's open questions are two independently posed problems: Conjecture 3 for every even $n\ge 28$, whose remaining open instances are $n=30$ and even $n>10000$, and Question 1, which given the proven results is open exactly at the orders $n\equiv 3\pmod 4$, $n\ge 11$.

Background

A latin square of order $n$ is an $n\times n$ array filled with $n$ symbols so that each symbol occurs exactly once in each row and each column. A transversal is a set of $n$ entries of such a square that includes each row, each column, and each symbol exactly once; two transversals are disjoint if they share no entry. Disjoint transversals have been central since Euler, because a partition of a square into $n$ disjoint transversals is exactly what is needed to build an orthogonal mate (a second latin square of the same order which, superimposed on the first, shows every ordered pair of symbols exactly once). A latin square is equivalent to a proper colouring of the edges of the complete bipartite graph $K_{n,n}$ using $n$ colours (rows and columns are the two vertex parts, symbols are colours), and in this language a transversal is a rainbow perfect matching: a perfect matching whose edges have pairwise distinct colours. For the history of the subject see the survey of Wanless (2011).

Transversal behaviour splits sharply by parity of the order. Ryser's conjecture asserts that every latin square of odd order has a transversal, and remains open. For even orders, Cavenagh and Wanless (2017) showed that there are superexponentially many latin squares with no transversal at all. In the opposite direction, Bowtell and Montgomery (2025) proved that a random latin square asymptotically almost surely admits a decomposition into disjoint transversals, so typical squares are extremely rich in disjoint transversals.

Two recent results exhibit squares that are pathologically far from this typical behaviour. Ghafari and Wanless (2024) constructed, for every even $n\ge 10$, a latin square with at least $\lfloor n/6\rfloor$ pinned entries, where an entry is called pinned if it is contained in every transversal of the square (and the square has at least one transversal). The source paper (Ghafari and Wanless, 2026) pushes further in two directions.

First, answering a question posed to the authors by Nick Wormald, it studies whether a square can have no two disjoint transversals while also having no pinned entry. The authors conjecture (their Conjecture 3) that for every even $n\ge 28$ such a square exists, and prove it for $n=28$ and all even $n$ with $32\le n\le 10000$. Their squares are small perturbations of the addition table of the cyclic group $\mathbb{Z}_n$, and the engine of the proof is a parity obstruction: writing $\Delta(r,c,s)=s-r-c\pmod n$, every transversal $T$ of a latin square of even order $n$ indexed by $\mathbb{Z}_n$ satisfies

$$\sum_{(r,c,s)\in T}\Delta(r,c,s)\equiv n/2\pmod n,$$

and in the constructed squares this forces every transversal to contain at least two entries of a fixed set of three entries, so any two transversals intersect by the pigeonhole principle. The absence of a pinned entry is then verified computationally up to $n=10000$ by a randomised hill-climbing algorithm that finds transversals avoiding any prescribed entry. The pairwise-intersection half of the construction works for every even $n\ge 28$ except $n=30$ (and Egan and Wanless (2012) showed no square with the full Conjecture 3 behaviour exists for $n\le 8$), but no general argument produces the required transversals beyond $n=10000$, and $n=30$ is not reached at all.

Second, the paper introduces the notion of a dominant transversal: a transversal that intersects every other transversal of the same square. Random squares asymptotically almost surely have none (an adaptation of the Bowtell–Montgomery machinery), group Cayley tables (the operation tables of finite groups) never have one (multiplying the symbols of a transversal through by a fixed group element produces pairwise disjoint transversals), and diagonally cyclic squares — squares admitting an automorphism that cycles rows, columns, and symbols simultaneously — never have one either. Nevertheless the authors prove (their Theorem 13) that a square with a dominant transversal exists for every $n>4$ with $n\not\equiv 3\pmod 4$ and $n\ne 6$, and exhibit one of order $7$; Egan and Wanless (2012) showed no square of order $6$ has one. The notion also bears on Ryser's conjecture: a dominant transversal in a square of order $n$ is equivalent to a proper $n$-colouring of the edges of $K_{n,n}$ with one perfect matching removed that has no rainbow perfect matching, so dominant transversals at odd orders push explicitly toward counterexample territory for Ryser's conjecture.

Problem Statement

The source's open questions are two independently posed problems, each asking whether an extreme transversal-intersection structure is realizable in a latin square of order $n$ at every order in its stated range.

(A) Pairwise-intersecting transversals with no pinned entry (Conjecture 3 of Ghafari and Wanless, and its frontier). Does there exist, for every even $n\ge 28$, a latin square of order $n$ with at least two transversals, no two of which are disjoint, and no pinned entry? The specifically open instances are: (i) $n=30$, where not even a latin square whose transversals pairwise intersect is known; (ii) even $n>10000$, where the known construction gives the pairwise-intersection property but the absence of a pinned entry has only been verified computationally up to $n=10000$. Orders outside even $n\ge 28$ are not part of the conjecture: the source remarks only speculatively that it may be true for some values of $n<28$ (although it fails for $n\le 8$) and that it would be most interesting to know if it can be true for any odd value of $n$; these remarks are context, not requirements of the source's questions.

(B) Dominant transversals (Question 1 of Ghafari and Wanless). Does every integer $n\ge 7$ — equivalently, since all other orders are settled, every $n\equiv 3\pmod 4$ with $n\ge 11$ — admit a latin square that has a dominant transversal? The answer may be negative: exhibiting a single order $n\ge 7$ for which no latin square has a dominant transversal would resolve the question, and the source authors report failed computer searches at orders $11$ and $15$.

(A) and (B) are two logically independent questions, posed separately in the source with no joint acceptance condition: a complete answer to either — a proof over its full stated range, or, for (B), a single order $n\ge 7$ at which no latin square has a dominant transversal — fully resolves that question, and neither requires the other to be resolved. Progress on individual listed instances within a question is meaningful partial progress on that question but does not close it. (The source paper also asks, informally, for a theoretical analysis of its heuristic transversal-finding algorithm; that methodological thread has no determinate acceptance condition and is not part of the completion boundary of this problem.)

References

1. Afsane Ghafari and Ian M. Wanless, Latin Squares whose transversals intersect in unusual ways, arXiv:2607.17547, submitted 20 July 2026, DOI 10.48550/arxiv.2607.17547, https://arxiv.org/abs/2607.17547 (source paper)
2. Afsane Ghafari and Ian M. Wanless, Latin Squares whose transversals share many entries, arXiv:2412.12466, submitted 17 December 2024, DOI 10.48550/arxiv.2412.12466, https://arxiv.org/abs/2412.12466
3. Candida Bowtell and Richard Montgomery, Almost every Latin square has a decomposition into transversals, arXiv:2501.05438, submitted 9 January 2025, DOI 10.48550/arxiv.2501.05438, https://arxiv.org/abs/2501.05438
4. Nicholas J. Cavenagh and Ian M. Wanless, Latin Squares with No Transversals, The Electronic Journal of Combinatorics 24(2) (2017), #P2.45, DOI 10.37236/6481, https://doi.org/10.37236/6481
5. Judith Egan and Ian M. Wanless, Latin Squares with Restricted Transversals, Journal of Combinatorial Designs 20 (2012), 344-361, DOI 10.1002/jcd.21316, https://doi.org/10.1002/jcd.21316
6. Ian M. Wanless, Transversals in latin squares: a survey, in Surveys in Combinatorics 2011, London Mathematical Society Lecture Note Series 392, Cambridge University Press (2011), 403-437, DOI 10.1017/cbo9781139004114.010, https://doi.org/10.1017/cbo9781139004114.010
7. Billy Child and Ian M. Wanless, Latin hypercubes with restricted transversals, arXiv:2605.01813, submitted 3 May 2026, DOI 10.48550/arxiv.2605.01813, https://arxiv.org/abs/2605.01813