All Questions
10 questions
3
votes
2
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197
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Proper Latin sub-squares of generalized Latin squares
Say we have a generalization of a Latin square, where the square is of size $n \times n$, $n = ab$ and each row and each column has $b$ occurrences of each of $[1, \dots, a]$. Is there always ...
- 341
1
vote
1
answer
118
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A bound on the number of partial transversals of a latin square
A Latin Square (LS) of order $n$ is an $n$ on $n$ matrix, each entry contains one of the symbols $1,2,\ldots,n$, and every row and every column contains each symbol exactly once. A (complete) ...
- 5,429
3
votes
1
answer
131
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A graph $G$ with two $K_6$ subgraphs, in which any one-factor of $G$ induces a one-factor in exactly one of the $K_6$ subgraphs?
I'm seeking a simple graph $G$ of the following type:
It contains two disjoint copies of $K_6$ (the complete graph on 6 nodes), $H$ and $H'$ say.
Any one-factor of $G$ must contain either (a) a one ...
- 32.1k
3
votes
1
answer
311
views
Evans conjecture for symmetric latin squares
The Evans conjecture ( which was proved later by Smetaniuk) states that for any $n$, if at most $n-1$ entries of a partial $n\times n$ latin square are filled, it can be completed to the full latin ...
- 188k
2
votes
2
answers
228
views
The edge precoloring extension problem for complete graphs
Consider coloring the edges of a complete graph on even order. This can be seen as the completion of an order $n$ symmetric Latin square except the leading diagonal. My question pertains to whether we ...
- 10.7k
2
votes
1
answer
167
views
graph built from orthogonal Latin Squares
I've asked the following question on MathExchange site, with a bounty, with no answer or comments. Maybe I would have additional comments here. The problem came to be while reading some articles on ...
- 4,589
5
votes
0
answers
105
views
Dinitz Conjecture extension to rectangles
The Dinitz Conjecture, which was proved later in a more general form by Galvin, stated that given an $n\times n$ array, its elements could be filled exactly like a latin square, where the elements in ...
- 2,089
3
votes
2
answers
414
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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 ...
- 4,229
4
votes
1
answer
534
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.
\[L=\left(...
13
votes
0
answers
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 ...
- 4,229