Rooted bicubic planar maps via Dyck paths

We provide a combinatorial proof of Tutte's decomposition of rooted bicubic planar maps into 3-connected components. Motivated by the framework of Bell transformations, we establish an explicit bijection between rooted bicubic planar maps on $2n$ vertices and Dyck paths of semilength $3n$ with ascents of length divisible by 3, where each $3j$-ascent is colored using one of $g_j$ colors corresponding to the rooted 3-connected bicubic maps on $2j$ vertices. Our bijection gives a constructive method for assembling all rooted bicubic planar maps from their 3-connected building blocks. We give a simple proof for the fact that every 3-connected bicubic planar map on $2n$ vertices with $n \geq 4$ can be obtained from a smaller primitive map through just two insertion operations that add either 4 or 6 vertices. Finally, we briefly discuss rootings of 3-connected bicubic maps, providing lower bounds on the minimal number of rootings and showing that prism graphs can be used in combination with our insertion operations to generate maps with the maximum of $6n$ distinct rootings for all $n \geq 11$.