The Maximum Number of Alternating Hexagonal Faces in (4,6)-fullerenes
The Maximum Number of Alternating Hexagonal Faces in (4,6)-fullerenes
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(4,6)-富勒烯中交替六边形面的最大数量
DOI：
10.11845/sxjz.2020180b
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发表时间：
2022
期刊：
影响因子：
--
通讯作者：
Heping Zhang
中科院分区：
文献类型：
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作者：
Lingjuan Shi;Heping Zhang
A(4,6)-富勒烯图G是一个只有方面和六边形面的连通平面三次图，是可能的硼氮富勒烯或非经典碳富勒烯的分子图。G的完美匹配或Kekule结构是覆盖G的所有顶点的不相交边的集合，G的交替集是G的面的集合，其边界是G的M个交替循环，对于G的完美匹配M，最大交替集的大小是Fries数。众所周知，六边形体系和(4,6)-富勒烯的弗里斯数分别等于它们的最大反强迫数（见[离散应用数学]）。[j] .数学学报，2017,33(2):187-194。考虑只包含(4,6)-富勒烯G中六边形面的交替集的最大尺寸是很自然的，这可以称为通常的Fries数。本文给出了一个仅依赖于G的阶数来计算G的通常Fries数的公式，并证明了G的通常Fries数为v(G)/3当且仅当G是跳跃式(4,6)-富勒烯。
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 Kekule 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.