Queue layouts and nonrepetitive colouring of planar graphs and powers of trees

Dujmovi\'{c}, Joret, Micek, Morin, Ueckerdt and Wood recently in [Planar graphs have bounded queue-number, Journal of the ACM, Volume 67, Issue 4, Article No.: 22, August 2020] showed some attractive graph product structure theorems for planar graphs. By using the product structure, they proved that planar graphs have bounded queue-number $48$; in [Planar graphs have bounded nonrepetitive chromatic number, Advances in Combinatorics, 5, 11 pp, 2020], the authors proved that planar graphs have bounded nonrepetitive chromatic number $768$. In this paper, still by using some product structure theorem, we improve the upper bound of queue-number of planar graphs to $27$ and the non-repetitive chromatic number to $320$. We also study powers of trees. We show a graph product structure theorem of the $k$-th power $T^k$ of tree $T$, then use it giving an upper bound of the nonrepetitive~chromatic~number of $T^k$. We also give an asymptotically tight upper bound of the queue-number of $T^k$.