Chen, Da Qi2016-08-292016-08-292016-08-292016-08-24http://hdl.handle.net/10012/10711Tutte's Four-Flow Conjecture states that every bridgeless, Petersen-free graph admits a nowhere-zero 4-flow. This hard conjecture has been open for over half a century with no significant progress in the first forty years. In the recent decades, Robertson, Thomas, Sanders and Seymour has proved the cubic version of this conjecture. Their strategy involved the study of the class of cyclically 5-connected cubic graphs. It turns out a minimum counterexample to the general Four-Flow Conjecture is also cyclically 5-connected. Motivated by this fact, we wish to find structural properties of this class in hopes of producing a list of minor-minimal cyclically 5-connected graphs.enGraph TheoryGraph FlowsStructural Graph TheoryMaster Thesis