We show that there ex­ists an ad­ja­cency la­belling scheme for pla­nar graphs where each ver­tex of an $n$-ver­tex pla­nar graph is as­signed a $(\log n+o(\log n))$-bit label. This is op­ti­mal up to the lower-order term. This re­sult ex­tends to a much larger class of graphs, namely any sub­graph of a strong prod­uct $H\boxtimes P$ where $H$ is a graph of bounded treewidth and $P$ is a path. This class in­cludes bounded-genus graphs and $k$-pla­nar graphs.