Exact distance coloring in trees

Abstract: For an integer $q\ge 2$ and an even integer $d$, consider the graph obtained from a large complete $q$-ary tree by connecting with an edge any two vertices at distance exactly $d$ in the tree. This graph has clique number $q+1$, and the purpose of this short note is to prove that its chromatic number is $\Theta\big(\tfrac{d \log q}{\log d}\big)$. It was not known that the chromatic number of this graph grows with $d$. As a simple corollary of our result, we give a negative answer to a problem of van den Heuvel and Naserasr, asking whether there is a constant $C$ such that for any odd integer $d$, any planar graph can be colored with at most $C$ colors such that any pair of vertices at distance exactly $d$ have distinct colors. Finally, we study interval coloring of trees (where vertices at distance at least $d$ and at most $cd$, for some real $c>1$, must be assigned distinct colors), giving a sharp upper bound in the case of bounded degree trees.