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A347941 For sets of n random points in the real plane, a(n) is an upper bound for the minimal number of nearest neighbors. 1
2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 4, 4, 4, 4, 4, 4, 5, 5, 5, 6, 6, 6, 6, 6, 6, 7, 7, 7, 8, 8, 8, 8, 8, 8, 9, 9, 9, 10, 10, 10, 10, 10, 10, 11, 11, 11, 12, 12, 12, 12, 12, 12, 13, 13, 13, 14, 14, 14, 14, 14, 14, 15, 15, 15, 16, 16, 16, 16, 16, 16, 17, 17, 17, 18, 18, 18, 18, 18, 18, 19, 19, 19, 20, 20, 20, 20, 20, 20, 21, 21, 21, 22, 22, 22, 22, 22, 22, 23 (list; graph; refs; listen; history; text; internal format)
OFFSET
2,1
COMMENTS
The sequence deals with sets of n points with pairwise different distances. The randomness in the definition provides for pairwise different distances with probability = 1.
A point A is called a nearest neighbor if there is a point B with smaller distance to A than to any other point C.
In graph theory terms: Let G be a simple digraph; the vertices of G are n arbitrarily placed points in R^2 with pairwise different distances; the edges of G are arrows joining each point (tail end) to its nearest neighbor (head end). Let b(n) be the minimal number of points receiving arrowheads in any such graph. a(n) is the best upper bound yet known for b(n).
A261953(n) for n >= 2 can be seen as an "inverse" to a(n).
a(n) is built by constructing G with n points and m nearest neighbors, m chosen as minimal as possible, then defining a(n)=m.
The start is a(n)=2 for n <= 9 and a(n)=3 for n=10,11,12. We call the pairs (n,m)=(9,2) and (n,m)=(12,3) "anchor pairs" and proceed to bigger n by combining graphs with these anchor pairs to bigger graphs. So the next anchor pairs are (18,4), (21,5) and (27,6).
If (n0,m-1) and (n1,m) are anchor pairs then a(n')=m for n0 < n' <= n1.
We conjecture that a(n) is optimal. This claim is true if the following assumptions hold:
- The anchor pairs (9,2) and (12,3) are optimal.
- All bigger anchor pairs (n,m) are constructed by combining copies of (9,2) if m is even and adding one (12,3) if m is odd.
LINKS
Manfred Boergens, Next-neighbours
FORMULA
a(2) = a(3) = 2.
a(n) = 2j for n = 9j-5 ... 9j, j > 0;
a(n) = 2j+1 for n = 9j+1 ... 9j+3, j > 0;
With h=(n+5)/9 for n>3:
a(n) = 2*floor(h) if h-floor(h)<2/3;
a(n) = 2*floor(h)+1 otherwise.
G.f.: -x^2*(x^11-2*x^9+x^8+2)/(-x^10+x^9+x-1). - Alois P. Heinz, Sep 20 2021
EXAMPLE
G with 25 vertices has at least 6 nearest neighbors (conjectured; it is proved that there are G with n=25 and m=6 but it is not yet proved that 6 is the minimum).
MATHEMATICA
h=(n+5)/9; Join[{2, 2}, Table[2 Floor[h] + If[FractionalPart[h]<2/3, 0, 1], {n, 4, 100}]]
CROSSREFS
Cf. A261953.
Sequence in context: A072746 A179528 A105390 * A013941 A061798 A345206
KEYWORD
nonn,easy
AUTHOR
Manfred Boergens, Sep 20 2021
STATUS
approved

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Last modified April 16 10:29 EDT 2024. Contains 371709 sequences. (Running on oeis4.)