Queen's Graph
Gordon Royle
gordon at csse.uwa.edu.au
Thu Sep 11 10:45:12 CEST 2003
Hi Seqfans..
This subject arose in GRAPHNET, which I am sure some of you also subscribe
to, but I think it is of interest to sequence fans..
Vasek Chvatal raised the (old) question of considering the number of colours
needed to colour the n x n "queen's graph" - the graph with vertex set equal
to the cells on a chessboard, with two vertices adjacent if a queen can move
from one to the other.
Clearly, as each row/column is an n-clique, this graph needs at least n
colours.
And in fact, when n congruent to 1,5 mod 6 it can be shown that n colours
suffices - denote this by (*).
For small values of n, we have
n=3 5 colours
n=4 5 colours
n=5 5 colours (*)
n=6 7 colours
n=7 7 colours (*)
n=8 9 colours
n=9 10 colours
n=10 11 colours
n=11 11 colours (*)
But then Guenter Stertenbrink supplied a 12-colouring of the 12x12 queens
graph
n = 12 12 colours
n = 13 13 colours (*)
Although there are many queen-related sequences in the OEIS, I cannot find
this one... so will add it in a few days after asking for any further
information and/or comments....
Cheers
gordon
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