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1
vote
1answer
89 views

Connections between loops (algebraic structure) and graphs

I would like to know whether there are known constructions which provide a bijection between loops (isomorphism classes) and (possibly directed) graphs. Any reference to a useful paper in this ...
6
votes
1answer
158 views

Is there a simple proof that there is no five mutually orthogonal Latin squares of order 6?

It is well known that there is a projective plane of order $n$ if and only if there exist a set of $n-1$ mutually orthogonal Latin squares. The first nontrivial case is $n=6$, which fails because of ...
7
votes
2answers
360 views

How many finite loops?

How many finite loops of order $n$ are there? I am interested in the exact values ​​of $n$ if $n <40$ or even reasonable estimates. I am also interested in formulae or bounds for all $n$. Note ...
2
votes
1answer
199 views

Are all symmetric idempotent Latin squares known?

Are all symmetric idempotent Latin squares known? There is such a square of order $n$ if and only if $n$ is odd. However, is there a classification of all of them? (The motivation for the question ...
10
votes
0answers
112 views

Proving that the set of $\lfloor n/3 \rfloor+1$ partial Latin squares given by Pebody is unavoidable?

Introduction Cutler and Öhman (2006) attribute to Pebody (via personal communication) a construction of a set of $k:=\lfloor n/3 \rfloor+1$ partial Latin squares which are unavoidable (i.e., any ...
4
votes
0answers
125 views

Existence problem for a generalisation of Latin squares (matrices with fixed row and column sets)

Let $R_1,\ldots,R_n$ and $C_1,\ldots,C_n$ be sets of size n. When does there exist an $n \times n$ matrix in which the $i$-th row is a permutation of $R_i$, for all $1 \leq i \leq n$, and the ...
3
votes
2answers
322 views

What is the number of k-regular subgraphs of $K_{12,12}$?

I'm thinking about making an attempt at counting the number of Latin squares of order 12. Currently I'm in (what I call) the "sanity check" stage, where we check that the ranges that need searching ...
7
votes
1answer
654 views

(0,1)-matrix congruence: is it known?

[[UPDATE: This work has now been published at SIAM J Discrete Math.: Formulae for the Alon–Tarsi Conjecture.]] By equating two formulae (one congruence by Glynn (1) (which has just appeared) and one ...
12
votes
0answers
1k views

Finding a chromatic polynomial by polynomial fitting

I would like to find the chromatic polynomial χ for the n by m rook's graph Gn,m for as many values of n and m possible. The rooks graph is also (a) the line graph of the complete bipartite graph ...
3
votes
1answer
394 views

Diagonally-cyclic Steiner Latin squares

A Steiner triple system is a decomposition of $K_n$ into $K_3$, such as $S=\{013,026,045,124,156,235,346\}$. Steiner triple systems give rise to a Steiner Latin squares, such as $L$ below. ...