OFFSET
3,1
LINKS
Andrew Baker and Joe Sawada, Magic Labelings on Cycles and Wheels, Lecture Notes in Computer Science 5165 (Combinatorial Optimization and Applications. Second International Conference, COCOA 2008). pp. 361-373.
Mukkai S. Krishnamoorthy, Allen Lavoie, Ali Dasdan, Bharath Santosh, Number of unique Edge-magic total labelings on Path P_n, arXiv:1402.2878 [math.CO], 2014.
EXAMPLE
From Gheorghe Coserea, May 23 2018: (Start)
For n=4 the a(4)=6 solutions are:
[1, 4, 8, 3, 2, 6, 5, 7]
[1, 5, 6, 4, 2, 7, 3, 8]
[1, 5, 8, 2, 4, 3, 7, 6]
[1, 7, 5, 2, 6, 3, 4, 8]
[3, 4, 8, 1, 6, 2, 7, 5]
[3, 6, 5, 1, 8, 2, 4, 7]
The solution [1, 4, 8, 3, 2, 6, 5, 7] is an encoding of the following vertex-magic labeling on C_4:
7 1 4 8 3 2 6
o-----o-----o-----o
| |
+-----------------+
5
In this labeling vertices are labeled 7, 4, 3, 6 while edges are labeled 1, 8, 2, 5 respectively. The vertex-magic constant of labeling k is 13 since k = 5+7+1 = 1+4+8 = 8+3+2 = 2+6+5.
In general, for C_n the magic constant of labeling k satisfies 3*n+1 - floor((n-1)/2) <= k <= 3*n+2 + floor((n-1)/2) and this bounds are tight for n>=6.
The solutions for n=4 have been generated using the MiniZinc model (e.g. $ minizinc -a -D"n=4;" magiccn.mzn | sort).
(End)
PROG
(MiniZinc)
% filename: magiccn.mzn : generate solution of size n
% usage: minizinc -a --soln-sep "" --search-complete-msg "" -D"n=5; " magiccn.mzn
include "globals.mzn";
int: n;
int: lo = 3*n+1 - (n-1) div 2;
int: hi = 3*n+2 + (n-1) div 2;
array[1..2*n] of var 1..2*n: x;
var lo..hi: h;
constraint alldifferent(x);
constraint forall([h = x[2*i-1] + x[2*i] + x[2*i+1] | i in 1..n-1]);
constraint h = x[2*n-1] + x[2*n] + x[1];
constraint forall([x[1] < x[2*i+1] | i in 1..n-1]); % break rotations
constraint x[2] < x[2*n]; % break reflection symmetry
solve satisfy;
output [show(x)];
% Gheorghe Coserea, May 22 2018
CROSSREFS
KEYWORD
nonn,more
AUTHOR
Andrew Baker (abaker04(AT)uoguelph.ca), Oct 16 2008
STATUS
approved