The tree-width of a graph G is the minimum k such that G may be decomposed into a "tree-structure" of pieces each with at most k + l vertices. We prove that this equals the maximum k such that there… Expand

A graph G is perfect if for every induced subgraph H, the chromatic number of H equals the size of the largest complete subgraph of H, and G is Berge if no induced subgraph of G is an odd cycle of… Expand

It is proved that every directed graph with no “haven” of large order has small tree-width, and the Hamilton cycle problem and other NP-hard problems can be solved in polynomial time when restricted to digraphs of bounded tree- width.Expand

Another proof is given, still using a computer, but simpler than Appel and Haken's in several respects, that every loopless planar graph admits a vertex-colouring with at most four different colours.Expand

Let G be an n-vertex graph with no minor isomorphic to an h- vertex complete graph. We prove that the vertices of G can be partitioned into three sets A, B, C such that no edge joins a vertex in A… Expand

It is shown (without assuming the 4CC) that every minimal counterexample to Hadwiger's conjecture whent=5 is “apex”, that is, it consists of a planar graph with one additional vertex.Expand

It is proved 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 graphs has aMinor isomorphism to the 2 k-spoke double wheel, the k-rung circular ladder, thek -rung Mobius ladder, or K 4, k.Expand