Bogdan Oporowski

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.

Typical subgraphs of 3- and 4-connected graphs

Abstract We prove that, for every positive integer k , there is an integer N such that every 3-connected graph with at least N vertices has a minor isomorphic to the k -spoke wheel or K 3, k ; and that every internally 4-connected graph with at least N vertices has a minor isomorphic to the 2 k -spoke double wheel, the k -rung circular ladder, the k -rung Mobius ladder, or K 4, k . We also prove an analogous result for infinite graphs.

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.

Some results on tree decomposition of graphs

We investigate tree decompositions (T,(Xt)tϵV(T)) whose width is “close to optimal” and such that all the subtrees of T induced by the vertices of the graph are “small.” We prove the existence of such decompositions for various interpretations of “close to optimal” and “small.” As a corollary of these results, we prove that the dilation of a graph is bounded by a logarithmic function of the congestion of the graph thereby settling a generalization of a conjecture of Bienstock.

On tree-partitions of graphs

A graph G admits a tree-partition of width k if its vertex set can be partitioned into sets of size at most k so that the graph obtained by identifying the vertices in each set of the partition, and then deleting loops and parallel edges, is a forest. In the paper, we characterize the classes of graphs (finite and infinite) of bounded tree-partition-width in terms of excluded topological minors.

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.

On infinite antichains of matroids

Abstract Robertson and Seymour have shown that there is no infinite set of graphs in which no member is a minor of another. By contrast, it is well known that the class of all matroids does contains such infinite antichains. However, for many classes of matroids, even the class of binary matroids, it is not known whether or not the class contains an infinite antichain. In this paper, we examine a class of matroids of relatively simple structure: M a,b,c consists of those matroids for which the deletion of some set of at most a elements and the contraction of some set of at most b elements results in a matroid in which every component has at most c elements. We determine precisely when M a,b,c contains an infinite antichain. We also show that, among the matroids representable over a finite fixed field, there is no infinite antichain in a fixed M a,b,c; nor is there an infinite antichain when the circuit size is bounded.