Enumeration of rooted 3-connected bipartite planar maps

Abstract

We provide the first solution to the problem of counting rooted 3-connected bipartite planar maps. Our starting point is the enumeration of bicoloured planar maps according to the number of edges and monochromatic edges, following Bernardi and Bousquet-Mélou [J. Comb. Theory Ser. B, 101 (2011), 315-377]. The decomposition of a map into 2- and 3-connected components allows us to obtain the generating functions of 2-and 3-connected bicoloured maps. Setting to zero the variable marking monochromatic edges we obtain the generating function of 3-connected bipartite maps, which is algebraic of degree 26. We deduce from it an asymptotic estimate for the number of 3-connected bipartite planar maps of the form $t \cdot n^{-5/2} γ^n$, where $γ=ρ^{-1} \approx 2.40958$ and $ρ\approx 0.41501$ is an algebraic number of degree 10.

Figures (5)

• Figure 1: The decomposition of a rooted planar map $\textsf{m}$ into its 2-core $\mathsf{C}(\textsf{m})$ (in red), i.e. the non-separable map containing the root, and maps attached by their root vertex to the corners of $\mathsf{C}(\textsf{m})$.
• Figure 2: Decomposition of a series map (left) and its dual parallel map (right) into smaller 2-connected maps. The series map is obtained from the two smaller maps by removing the two root edges, identifying vertex $a$ with $c$, and adding a root edge from $d$ to $b$. The (dual) parallel map is obtained from the two smaller maps by identifying vertex $a$ with $c$ and vertex $b$ with $d$ then removing the edge $cd$.
• Figure 3: Decomposition of a polyhedral map $\textsf{m}$ into its 3-core $\mathsf{T}(\textsf{m})$ (in red) and smaller 2-connected maps.
• Figure 4: The smallest 3-connected bipartite planar maps pictured with their rootings and number of edges. The cube is the unique (unrooted) 3-connected bipartite map with 12 edges; it has 24 symmetries and thus admits a unique rooting. There are no 3-connected bipartite maps with 13, 14 and 15 edges. The unique 3-connected bipartite map with 16 edges has 8 symmetries, hence $[z^{16}]T_b(z) = 2\cdot 16/8 = 4$. There are two 3-connected bipartite maps with 18 edges, the hexagonal prism (left) has 12 symmetries and the other one has 6. The unique 3-connected bipartite map with 19 edges has only 2 symmetries. The three 3-connected bipartite maps with 20 edges admit 8 (left), 10 (middle) and 2 (right) symmetries. The rootings of the two maps with only 2 symmetries (middle right and bottom right) are not drawn for readability.
• Figure 5: The eight different 2-colourings of the rooted triangle map $\textsf{m}$, that are encoded by the Ising polynomial $P_{\textsf{m}}(2,\nu)$, and their decompositions. On the left are the 2-colourings of $\textsf{m}$ for which the root edge is monochromatic. Their contribution to $P_{\textsf{m}}(2,\nu)$ is $2\nu^3 + 2\nu$. Alongside them are the 2-coloured maps obtained after contracting (left) or deleting (right) their root edge. On the right are the 2-colourings of $\textsf{m}$ for which the root edge is bichromatic. Their contribution to $P_{\textsf{m}}(2,\nu)$ is $4\nu$. Alongside them are the 2-coloured maps obtained after deleting their root edge (recall that we do not contract a bichromatic root edge). Thus only the left side contributes to $P_{\textsf{m}_/}(2,\nu) = 2\nu^2 + 2$, while both sides contribute to $P_{\textsf{m}_{\backslash}}(2,\nu) = 2\nu^2 + 4\nu + 2$. Observe that we indeed have $P_{\textsf{m}}(2,\nu) = P_{\textsf{m}_{\backslash}}(2,\nu) + (\nu - 1)P_{\textsf{m}_/}(2,\nu) = 2\nu^3 + 6\nu$.

Theorems & Definitions (2)

• Theorem 1
• Theorem 2