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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)
This article proves the conjecture of Thomas that, for every graph G, there is an integer k such that every graph with no minor isomorphic to G has a 2-coloring of either its vertices or its edges where each color induces a graph of tree-width at most k. Some generalizations are also proved.
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 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 MOE of M by an element xOE such that {x, xOE} is independent and MOE is unaltered by swapping the labels on x and xOE. When x is fixed, a representation of M 0 x extends in at most(More)
This paper introduces a generalization of the matroid operation of ∆ − Y exchange. This new operation, segment-cosegment exchange, replaces a coindependent set of k collinear points in a matroid by an independent set of k points that are collinear in the dual of the resulting matroid. The main theorem of the first half of the paper is that, for every field,(More)
Let F be a field and let N be a matroid in a class N N of F-representable matroids that is closed under minors and the taking of duals. Then N is an F-stabilizer for N N if every representation of a 3-connected member of N N is determined up to elementary row operations and column scaling by a representation of any one of its N-minors. The study of(More)