dc.contributor.author de Jong, Jamie dc.date.accessioned 2020-05-05 18:19:17 (GMT) dc.date.available 2020-05-05 18:19:17 (GMT) dc.date.issued 2020-05-05 dc.date.submitted 2020-04-30 dc.identifier.uri http://hdl.handle.net/10012/15821 dc.description.abstract In 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. en 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. dc.language.iso en en dc.publisher University of Waterloo en dc.subject.lcsh Graph theory en dc.title Jaeger’s Strong 3-Flow Conjecture for Graphs in Low Genus Surfaces en dc.type Doctoral Thesis en dc.pending false uws-etd.degree.department Combinatorics and Optimization en uws-etd.degree.discipline Combinatorics and Optimization en uws-etd.degree.grantor University of Waterloo en uws-etd.degree Doctor of Philosophy en uws.contributor.advisor Richter, Bruce uws.contributor.affiliation1 Faculty of Mathematics en uws.published.city Waterloo en uws.published.country Canada en uws.published.province Ontario en uws.typeOfResource Text en uws.peerReviewStatus Unreviewed en uws.scholarLevel Graduate en
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