What's the optimal bound of this graph coloring problem? Let $G$ be a graph with nodes $v_1, ..., v_n$ and a set of edges $E$. We denote $\chi(G)$ the chromatic number of G (smallest number of colors to color the graph such that an edge never connects two nodes with the same color) and $ω(G)$ the clique number (number of vertices of a maximal clique).
Let $x_1, ..., x_n$ be real numbers and we denote $\Delta = \max_{i, j} x_i - x_j$ the diameter of the $(x_i)_i$.
Let $M>0$ and we define the following constraints :
$$(v_i, v_j)\in E \Rightarrow |x_i-x_j|\geq M$$
I want to prove (or find a counter-example) that $$\Delta \geq (\chi(G) - 1)\times M$$
I already know how to prove that $\Delta \geq (ω(G) - 1)\times M$ but it's a weaker result.
As a small example, you can think about the graph with 5 vertices which form a ring : $\forall i, (v_i, v_{i+1})\in E$ which is a minimal example of a graph having $ω(G) < \chi(G)$.

Your graphs are the complement graphs of interval graphs (with interval centered around $x_i$ and of size $M$). These graphs are comparability graphs and are known to be perfect graphs (their chromatic number equals their clique number).
Also, note that a five cycle can't be represented as a comparability graph, as each odd cycle of length greater than 3 must have chords, so it is not a counterexample in your problem.
