Dirk Vertigan

Excluding any graph as a minor allows a low tree-width 2-coloring

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-widlh at most k. Some generalizations are also proved.

The Computational Complexity of the Tutte Plane: the Bipartite Case

Along different curves and at different points of the ( x, y )-plane the Tutte polynomial evaluates a wide range of quantities. Some of these, such as the number of spanning trees of a graph and the partition function of the planar Ising model, can be computed in polynomial time, others are # P -hard. Here we give a complete characterisation of which points and curves are easy/hard in the bipartite case.

Reseaux électriques planaires II.

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Partitioning into graphs with only small components

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 at most 57 vertices. Some generalizations of the last result are also discussed.

On Inequivalent Representations of Matroids over Finite Fields

Kahn conjectured in 1988 that, for each prime powerq, there is an integern(q) such that no 3-connectedGF(q)-representable matroid has more thann(q) inequivalentGF(q)-representations. At the time, this conjecture was known to be true forq=2 andq=3, and Kahn had just proved it forq=4. In this paper, we prove the conjecture forq=5, showing that 6 is a sharp value forn(5). Moreover, we also show that the conjecture is false for all larger values ofq.

Surfaces, Tree-Width, Clique-Minors, and Partitions

In 1971, Chartrand, Geller, and Hedetniemi conjectured that the edge set of a planar graph may be partitioned into two subsets, each of which induces an outerplanar graph. Some partial results towards this conjecture are presented. One such result, in which a planar graph may be thus edge partitioned into two series-parallel graphs, has nice generalizations for graphs embedded onto an arbitrary surface and graphs with no large clique-minor. Several open questions are raised.

Unavoidable Minors of Large 3-Connected Matroids

This paper proves that, for every integernexceeding two, there is a numberN(n) such that every 3-connected matroid with at leastN(n) elements has a minor that is isomorphic to one of the following matroids: an (n+2)-point line or its dual, the cycle or cocycle matroid ofK3,n, the cycle matroid of a wheel withnspokes, a whirl of rankn, or ann-spike. A matroid is of the last type if it has ranknand consists ofnthree-point lines through a common point such that, for allkin {1,2,?,n?1}, the union of every set ofkof these lines has rankk+1.

Unavoidable Minors of Large 3-Connected Binary Matroids

We show that, for every integerngreater than two, there is a numberNsuch that every 3-connected binary matroid with at leastNelements has a minor that is isomorphic to the cycle matroid ofK3,n, its dual, the cycle matroid of the wheel withnspokes, or the vector matroid of the binary matrix (In|Jn?In), whereJnis then×nmatrix of all ones.

PARTITIONING GRAPHS OF BOUNDED TREE-WIDTH

The paper discusses vertex partitions and edge partitions of graphs of bounded tree-width into graphs of smaller tree-width. The rst part of the paper proves the existence of several kinds of such partitions. The second part, which has a Ramsey-theoretic character, shows that some of the results of the rst part are close to being best possible. The last section of the paper presents a result on partitioning graphs of bounded tree-width into star-forests.

Totally Free Expansions of Matroids

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 M? of M by an element x? such that {x, x?} is independent and M? is unaltered by swapping the labels on x and x?. When x is fixed, a representation of M\x extends in at most one way to a representation of M. A 3-connected matroid N is totally free if neither N nor its dual has a fixed element whose deletion is a series extension of a 3-connected matroid. The significance of such matroids derives from the theorem, established here, that the number of inequivalent representations of a 3-connected matroid M over a finite field F is bounded above by the maximum, over all totally free minors N of M, of the number of inequivalent F -representations of N. It is proved that, within a class of matroids that is closed under minors and duality, the totally free matroids can be found by an inductive search. Such a search is employed to show that, for all r?4, there are unique and easily described rank-r quaternary and quinternary matroids, the first being the free spike. Finally, Seymours Splitter Theorem is extended by showing that the sequence of 3-connected matroids from a matroid M to a minor N, whose existence is guaranteed by the theorem, may be chosen so that all deletions and contractions of fixed and cofixed elements occur in the initial segment of the sequence.