There are many hard problems in graph theory, such as the Four Color Problem, which can besolved in the general case if they are solvable for cubic graphs. Possible minimal counterexamplesfor most of these problems are asked to be snarks. An effective approach to prove theorems for snarks is by measures of edge-uncolorability. The thesis develops theory of cores and the related measure 3 for cubic graphs, first introduced by Steffen. The theory is applied to prove further results to both Fan-Raspaud conjecture and Petersen coloring conjecture, for which there are very few results known. Relations between 3 and other measures, especially the oddness, are given. Moreover, the theory of cores are extended to weak cores, which are then related to Fano-flows. It is also extend to cores of r-graphs. This gives us two benefits, one is to pose the generalized Fan-Raspaud conjecture from the viewpoint of cores, and the other is to define the rst measure for r-graphs. Finally, the thesis proves partial results to the generalized Berge conjecture and to Vizing's planar graph conjecture.