Distributions of Near-4-Regular Maps on the Sphere and the Projective Plane

Abstract

Let ${{{\mathcal {U}}}}_n$ and ${\overline{{{\mathcal {U}}}}}_n$ (${{{\mathcal {A}}}}_n$ and ${\overline{{{\mathcal {A}}}}}_n$) denote the sets of all rooted near-4-regular maps with n inner faces (n edges) on the sphere and the projective plane, respectively. Let $p_m$ and $\overline{p_m}$ be, respectively, the limit probability (as $n\rightarrow \infty$) of the event that the rooted vertex of a map chosen in ${{{\mathcal {U}}}}_n$ and ${\overline{{{\mathcal {U}}}}}_n$ (${{{\mathcal {A}}}}_n$ and ${\overline{{{\mathcal {A}}}}}_n$) at random is of valency 2m. It is shown that both $p_m$ and $\overline{p_m}$ obey the asymptotic pattern characterized by the factor $m^{\frac{1}{2}}$: $Cm^{\frac{1}{2}} {\left( \frac{2}{3}\right) }^m$ as $m\rightarrow \infty$, where C is a constant depending on the type of maps, meanwhile each of $q_m$ and $\overline{q_m}$ will not satisfy the root-vertex valency distribution pattern posed in Liskovets (J Combin Theory Ser B 75, 116–133, 1999) (i.e., $q_m=\overline{q_m}=0$ for every natural number m). In particular, those maps can not satisfy several other classical patterns for n-edged maps.