(4,6)-富勒烯图的最大交错六边形面数

The Maximum Number of Alternating Hexagonal Faces in(4,6)-fullerenes

(4,6)-富勒烯图G是一个连通平面3-正则图,它的每个面是四边形或者六边形;它是硼氮富勒烯或者非经典碳富勒烯的分子图.图G的一个完美匹配或者凯库勒结构是覆盖G的所有顶点的一个不交边的集合.如果存在图G的一个完美匹配M,使得G中一些面的边界是M-交错圈,那么这些面的集合称为G的一个交错集.一个最大交错集的大小称为Fries数.我们已经知道六角系统和(4,6)-富勒烯图的Fries数都等于其最大反强迫数(见[Discrete Appl.Math.,2016,202:95-105]和[Discrete Appl.Math.,2017,233:187-194]).接下来考虑(4,6)-富勒烯图中仅含六边形面的最大交错集的元素个数便是一个很自然的问题,该不变量称为通常Fries数.本文得到了一个计算图G的通常Fries数的公式,该公式仅与图顶点数相关.我们进一步证明了G的通常Fries数等于顶点数的三分之一当且仅当G是一个leapfrog(4,6)-富勒烯图.

A(4,6)-fullerene graph G is a connected plane cubic graph with only square and hexagonal faces,which is the molecular graph of possible boron-nitrogen fullerene or non-classical carbon fullerene.A perfect matching or a Kekulé structure of G is a set of disjoint edges covering all vertices of G.An alternating set of G is a set of faces of G whose boundaries are M-alternating cycles for a perfect matching M of G.The size of a maximum alternating set is the Fries number.It is known that the Fries numbers of hexagonal systems and(4,6)-fullerenes are equal to their maximum anti-forcing numbers,respectively(see[Discrete Appl.Math.,2016,202:95-105]and[Discrete Appl.Math.,2017,233:187-194]).It is natural to consider the maximum size of alternating sets only including hexagonal faces in a(4,6)-fullerene G,which may be called the usual Fries number.In this paper we obtain a formula only depending on its order to count the usual Fries number of G.We also show that the usual Fries number of G is v(G)/3 if and only if G is a leapfrog(4,6)-fullerene.