Guoli Ding

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.

Subgraphs and well-quasi-ordering

Let be a class of graphs and let ⪯ be the subgraph or the induced subgraph relation. We call ⪯ an ideal (with respect to ⪯) if ⪯ implies that ⪯. In this paper, we study the ideals that are well-quasiordered by ⪯. The following are our main results. If ⪯ is the subgraph relation, we characterize the well-quasi-ordered ideals in terms of exluding subgraphs. If⪯is the induced subgraph relation, we present three wellquasi-ordered ideals. We also construct examples to disprove some of the possible generalizations of our results. The connections between some of our results and digraphs are considered in this paper too.

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.

Bounding the vertex cover number of a hypergraph

AbstractFor a hypergraphH, we denote by(i)τ(H) the minimumk such that some set ofk vertices meets all the edges,(ii)ν(H) the maximumk such that somek edges are pairwise disjoint, and(iii)λ(H) the maximumk≥2 such that the incidence matrix ofH has as a submatrix the transpose of the incidence matrix of the complete graphKk. We show that τ(H) is bounded above by a function of ν(H) and λ(H), and indeed that if λ(H) is bounded by a constant then τ(H) is at most a polynomial function of ν(H).