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Let % be a class of graphs and let i be the subgraph or the induced subgraph relation. We call % an idea/ (with respect to I) if G I G' E % implies that G E %. In this paper, we study the ideals that are well-quasiordered by I. The following are our main results. If 5 is the subgraph relation, w e characterize the well-quasi-ordered ideals in terms of excluding subgraphs. If I 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. 01992

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. r

We show that, for every integer n greater than two, there is a number N such that every 3-connected binary matroid with at least N elements has a minor that is isomorphic to the cycle matroid of K 3, n , its dual, the cycle matroid of the wheel with n spokes, or the vector matroid of the binary matrix (I n | J n &I n ), where J n is the n_n matrix of all ones.

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