Problem

Source: Turkey TST 2019 Day 2 P6

Tags: combinatorics, graph theory



$k$ is a positive integer, $R_{n}={-k, -(k-1),..., -1, 1,..., k-1, k}$ for $n=2k$ $R_{n}={-k, -(k-1),..., -1, 0, 1,..., k-1, k}$ 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 $R_{n}$ 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 $R_{n}$ 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. $n\geq{3}$, if every mechanism that is labeled nicely with $R_{n}$, could be labeled precisely with $R_{m}$, what is the minimal value of $m$?