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Even Cycle and Even Cut Matroids
In this thesis we consider two classes of binary matroids, even cycle matroids and even cut matroids. They are a generalization of graphic and cographic matroids respectively. We focus on two mainExpand
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When does a biased graph come from a group labelling?
A biased graph consists of a graph G together with a collection of distinguished cycles of G, called balanced, with the property that no theta subgraph contains exactly two balanced cycles. PerhapsExpand
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Biased graphs with no two vertex-disjoint unbalanced cycles
Lov\'asz has completely characterised the structure of graphs with no two vertex-disjoint cycles, while Slilaty has given a structural characterisation of graphs with no two vertex-disjoint oddExpand
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Packing Steiner trees
Let T be a distinguished subset of vertices in a graph G. A T-Steiner tree is a subgraph of G that is a tree and that spans T. Kriesell conjectured that G contains k pairwise edge-disjoint T-SteinerExpand
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Relationships between Pairs of Representations of Signed Binary Matroids
We show how pairs of signed graphs with the same even cycles relate to pairs of grafts with the same even cuts. These results are proved in the more general context of signed binary matroids.
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Highly-connected planar cubic graphs with few or many Hamilton cycles
In this paper we consider the number of Hamilton cycles in planar cubic graphs of high cyclic edge-connectivity, answering two questions raised by Chia and Thomassen ("On the number of longest andExpand
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3-Flows with large support
We prove that every 3-edge-connected graph $G$ has a 3-flow $\phi$ with the property that $|\mathop{supp}(\phi)| \ge \frac{5}{6} |E(G)|$. The graph $K_4$ demonstrates that this $\frac{5}{6}$ ratio isExpand
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Highly contrasted responses of Mediterranean octocorals to climate change along a depth gradient
Climate change has a strong impact on marine ecosystems, including temperate species. Analysing the diversity of thermotolerance levels within species along with their genetic structure enables aExpand
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Stabilizer theorems for even cycle matroids
A signed graph is a representation of an even cycle matroid M if the cycles of M correspond to the even cycles of that signed graph. Two long standing open questions regarding even cycle matroids areExpand
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Maximum Size Binary Matroids with no AG(3, 2)-Minor are Graphic
We prove that the maximum size of a simple binary matroid of rank $r \geq 5$ with no $AG(3,2)$-minor is $\binom{r+1}{2}$ and characterize those matroids achieving this bound. When $r \geq 6$, the g...
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