平面与射影平面上近-4-正则地图的分布

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

设Un和Un(An和An)分别是平面和射影平面上具有n个内部面(n条边)的有根近-4-正则地图的全体．设pm与Pm分别表示n→∞时,从Un和Un(An和An)中随机选取一个地图使得它的根次为2m这一事件的概率极限．本文证明了如下结果:pm与Pm都遵从渐进模式m1/2:Cm1/2(2/3)m,当m→∞,其中C是与所选取地图类型有关的常数．同时,qm与qm都不服从Liskovets提出的关于地图根节点的次的分布规律(即对于每一个自然数m,qm=qm=0)．特别地,这几类地图都不满足其它几个关于地图分布的模式．

Let Un and Un （An and .An） be, respectively, the sets of rooted near-4- regular maps with n inner faces （n edges） on the plane and the projective plane. Let Pm and Pm be, respectively, the limit probabilities （as n→∞） of the eveices that the root vertex of a map chosen in Un and Un （An and An） at random are of valency 2m. It is shown that both Pm and Pm obey the asymptotic pattern characterized by the factor m^1/2：Cm^1/2 （2/3）^m as m →∞）, where C is a constant depending on the type of maps, meanwhile, each of qm and qm will not satisfy the root vertex valency distribution pattern posed by Liskovets （i.e., qm = qm = 0 for every natural number m）. In particular, those maps can not satisfy several other classical patterns for n-edged maps.