CN-TST 2019 Problem 18

Given positive integers $d \ge 3$ , $r>2$ and $l$ , with $2d \le l <rd$ . Every vertice of the graph $G(V,E)$ is assigned to a positive integer in $\{1,2,\cdots,l\}$ , such that for any two consecutive vertices in the graph, the integers they are assigned to, respectively, have difference no less than $d$ , and no more than $l-d$ . A proper coloring of the graph is a coloring of the vertices, such that any two consecutive vertices are not the same color. It's given that there exist a proper subset $A$ of $V$ , such that for $G$ 's any proper coloring with $r-1$ colors, and for an arbitrary color $C$ , either all numbers in color $C$ appear in $A$ , or none of the numbers in color $C$ appear in $A$ . Show that $G$ has a proper coloring within $r-1$ colors.