k is a positive integer,
Rn=−k,−(k−1),...,−1,1,...,k−1,k for n=2k
Rn=−k,−(k−1),...,−1,0,1,...,k−1,k for n=2k+1.
A mechanism consists of some marbles and white/red ropes that connects some marble pairs. If each one of the marbles are written on some numbers from Rn with the property that any two connected marbles have different numbers on them, we call it nice labeling. If each one of the marbles are written on some numbers from Rn with the properties that any two connected marbles with a white rope have different numbers on them and any two connected marbles with a red rope have two numbers with sum not equal to 0, we call it precise labeling.
n≥3, if every mechanism that is labeled nicely with Rn, could be labeled precisely with Rm, what is the minimal value of m? combinatoricsgraph theory