Dirk L. Vertigan

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Kahn conjectured in 1988 that, for each prime power q, there is an integer n(q) such that no 3-connected GF(q)-representable matroid has more than n(q) inequivalent GF(q)-representations. At the time, this conjecture was known to be true for q=2 and q=3, and Kahn had just proved it for q=4. In this paper, we prove the conjecture for q=5, showing that 6 is a(More)
The paper presents several results on edge partitions and vertex partitions of graphs into graphs with bounded size components. We show that every graph of bounded tree-width and bounded maximum degree admits such partitions. We also show that an arbitrary graph of maximum degree four has a vertex partition into two graphs, each of which has components on(More)
The aim of this paper is to give insight into the behaviour of inequivalent representations of 3-connected matroids. An element x of a matroid M is fixed if there is no extension MŒ of M by an element xŒ such that {x, xŒ} is independent and MŒ is unaltered by swapping the labels on x and xŒ. When x is fixed, a representation of M0x extends in at most one(More)