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.

让\({{{\mathcal {U}}}}_n\)和\({\overline{{{\mathcal {U}}}}}}_n\) ( \({{{\mathcal {A}} }}_n\)和\({\overline{{{\mathcal {A}}}}}_n\) ) 表示球体上具有n 个内面（n 个边）的所有有根近 4-正则映射的集合和投影平面，分别。让\（p_m \）和\（\划线{p_m} \）是分别限制概率（如\（N \ RIGHTARROW \ infty \）的情况下的），其在选择了一个图的根顶点\（{ {{\mathcal {U}}}}_n\)和\({\overline{{{\mathcal {U}}}}}_n\) ( \({{{\mathcal {A}}}}}_n\ )和\({\overline{{{\mathcal {A}}}}}_n\) ) 随机的化合价为 2 m。结果表明，\(p_m\)和\(\overline{p_m}\) 都服从由因子\(m^{\frac{1}{2}}\)表征的渐近模式：  \(Cm^{ \frac{1}{2}} {\left( \frac{2}{3}\right) }^m\)为\(m\rightarrow \infty\)，其中C是一个常数，取决于映射，同时\(q_m\)和\(\overline{q_m}\) 中的每一个都不会满足 Liskovets (J Combin Theory Ser B 75, 116–133, 1999) 中提出的根顶点价分布模式（即，\(q_m=\overline{q_m}=0\)对于每个自然数m）。特别是，这些地图不能满足n边地图的其他几种经典模式。