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dc.contributor.authorde Jong, Jamie 18:19:17 (GMT) 18:19:17 (GMT)
dc.description.abstractIn 1972, Tutte posed the 3-Flow Conjecture: that all 4-edge-connected graphs have a nowhere zero 3-flow. This was extended by Jaeger et al. (1992) to allow vertices to have a prescribed, possibly non-zero difference (modulo 3) between the inflow and outflow. He conjectured that all 5-edge-connected graphs with a valid prescription function have a nowhere zero 3-flow meeting that prescription. Kochol (2001) showed that replacing 4-edge-connected with 5-edge-connected would suffice to prove the 3-Flow Conjecture and Lovàsz et al. (2013) showed that both conjectures hold if the edge connectivity condition is relaxed to 6-edge-connected. Both problems are still open for 5-edge-connected graphs. The 3-Flow Conjecture was known to hold for planar graphs, as it is the dual of Grötzsch's Colouring Theorem. Steinberg and Younger (1989) provided the first direct proof using flows for planar graphs, as well as a proof for projective planar graphs. Richter et al. (2016) provided the first direct proof using flows of Jaeger's Strong 3-Flow Conjecture for planar graphs. We extend their result to graphs embedded in the projective plane. Lai (2007) showed that Jaeger's Strong 3-Flow Conjecture cannot be extended to 4-edge-connected graphs by constructing an infinite family of 4-edge-connected graphs that do not have a nowhere zero 3-flow meeting their prescribed net flow. We prove that graphs with arbitrarily many non-crossing 4-edge-cuts sufficiently far apart have a nowhere zero 3-flow, regardless of their prescription function. This is a step toward answering the question of which 4-edge-connected graphs have this property.en
dc.publisherUniversity of Waterlooen
dc.subject.lcshGraph theoryen
dc.titleJaeger’s Strong 3-Flow Conjecture for Graphs in Low Genus Surfacesen
dc.typeDoctoral Thesisen
dc.pendingfalse and Optimizationen and Optimizationen of Waterlooen
uws-etd.degreeDoctor of Philosophyen
uws.contributor.advisorRichter, Bruce
uws.contributor.affiliation1Faculty of Mathematicsen

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