On the chromatic number of powers of subdivisions of graphs

For a given graph $G=(V,E)$, we define its $n$th subdivision as the graph obtained from $G$ by replacing every edge by a path of length $n$. We also define the $m$th power of $G$ as the graph on vertex set $V$ where we connect every pair of vertices at distance at most $m$ in $G$. In this paper, we study the chromatic number of powers of subdivisions of graphs and resolve the case $m=n$ asymptotically. In particular, our result confirms a conjecture of Mozafari-Nia and Iradmusa in the case $m=n=3$ in a strong sense.