(4,5,6)-富勒烯图的共振圈、匹配强迫和反强迫的问题研究

匹配理论是图论及其应用中的一个重要分支。此外，（4,5,6）-富勒烯图包含了所有的（4,6）-和（5,6）-富勒烯图。因此，研究（4,5,6）-富勒烯有关匹配特性的问题，不仅促进匹配理论自身的发展，而且极大的丰富了富勒烯在数学方面的理论研究，为富勒烯在物理、化学以及医药方面的等领域的发展及应用提供有力的工具。. 本项目将对（4,5,6）-富勒烯图中的若干完美匹配特性的问题进行研究，以期解决一些具有挑战性的匹配问题。研究内容主要包括：(1)探究（4,5,6）-富勒烯图中共振的8-长圈和10-长圈；(2)研究（4,5,6）-富勒烯图最小强迫数；(3)探索（4,5,6）-富勒烯图最大强迫数和最大反强迫数的非平凡上界；(4)探讨以上问题中相应极值图的刻画。本项目所研究的问题进一步深化了有关图的匹配问题的理论研究，问题的解决进一步推动了匹配理论的发展。

Matching Theory is an important branch of Graph Theory and Its Applications. Furthermore, (4,5,6)-fullerenes include all (4,6)- and (5,6)-fullerenes. Hence, studying the problems about matching properties not only promotes the development for matching theory, but also greatly enriches the theoretical research of fullerene in mathematics and affords a powerful tool for the development and application of fullerenes in pthysics,chemistry and medicine, ect.. In this project we mainly study some perfect matching properties of (4,5,6)-fullerenes, and aim to solve some challenging problems in matchings. The main content of this project contains the following: Firstly, explore the resonant 8- and 10-cycles. Secondly, study the minimum forcing number of (4,5,6)-fullerenes. Thirdly, explore the nontrivial upper bounds of maximum forcing numbers and maximum anti-forcing numbers of (4,5,6)-fullerenes. Finally, explore the characterization of the corresponding extremal graphs in the above problems.