Question 54
For positive integer $k$, let
$$
R_n = \{-k, -(k-1), \dots, -1, 1, \dots, k-1, k\} \text{ for } n = 2k \text{ and}
$$
$$
R_n = \{-k, -(k-1), \dots, -1, 0, 1, \dots, k-1, k\} \text{ for } n = 2k+1.
$$
A device consists of several balls and red or white ropes connecting some ball pairs. A *labeling* is a coloring of each ball by one of the elements of $R_n$. We say that a labeling is *good* if colors of any two connected balls are different. We say that a labeling is *sensitive* if the colors of any two balls connected by white rope are different and the sum of colors of any two balls connected by red rope is not equal to $0$.
Let $n \ge 3$ be fixed. Suppose that any device which has a good labeling by $R_n$ has also a sensitive labeling by $R_m$. Find the smallest possible value of $m = m(n)$.
$$
R_n = \{-k, -(k-1), \dots, -1, 1, \dots, k-1, k\} \text{ for } n = 2k \text{ and}
$$
$$
R_n = \{-k, -(k-1), \dots, -1, 0, 1, \dots, k-1, k\} \text{ for } n = 2k+1.
$$
A device consists of several balls and red or white ropes connecting some ball pairs. A *labeling* is a coloring of each ball by one of the elements of $R_n$. We say that a labeling is *good* if colors of any two connected balls are different. We say that a labeling is *sensitive* if the colors of any two balls connected by white rope are different and the sum of colors of any two balls connected by red rope is not equal to $0$.
Let $n \ge 3$ be fixed. Suppose that any device which has a good labeling by $R_n$ has also a sensitive labeling by $R_m$. Find the smallest possible value of $m = m(n)$.