在图的完美匹配理论中，Fulkerson猜想是其中的一个核心猜想。Fulkerson猜想断言：每个无桥的3-正则图都有六个完美匹配，使得每条边包含在其中的恰好两个完美匹配上。3-正则图的核理论是近些年引入的一种新的研究方法。本项目将应用3-正则图的核理论来研究Fulkerson猜想及其相关联的另外三个重要猜想：Petersen着色猜想、Berge猜想和Fan-Raspaud猜想。目前，这四个猜想仅在非常有限的图类上得到了验证。本项目将考虑一个与Petersen着色猜想相关的问题，期望得到该问题的一个肯定的解答。对于其余三个猜想，本项目将通过核给出一些更为宽泛的图类，并论证猜想在这些图类上是成立的。一方面，本项目运用3-正则图的核理论来推动对以上四个猜想的研究；另一方面，在研究过程中，核理论本身也将得到丰富。

In the theory on perfect matchings of graphs, Fulkerson Conjecture is a conjecture in the central place. Fulkerson Conjecture states that every bridgeless cubic graph has six perfect matchings such that each edge is contained in precisely two of them. The theory on cores of cubic graphs was introduced recently. It provides a new way to consider some problems. In this project, we will apply the theory on cores to study Fulkerson Conjecture and three related conjectures, including Petersen Coloring Conjecture, Berge Conjecture and Fan-Raspaud Conjecture. So far, these four conjectures were verified for only a few classes of cubic graphs. First, we will consider a question related to the Petersen coloring conjecture, and expect to give a positive answer to this question. Then, for each remaining conjecture, we will search for a more relaxed condition in terms of cores under which this conjecture holds. On one hand, this project applies the theory on cores of cubic graphs to study these four conjectures. On the other hand, this study will enrich the theory on cores itself.