Configurations in Graphs of Large Minimum Degree, Connectivity, or Chromatic Number

Wagner [42] proved that a graph of large chromatic number contains a subgraph that can be contracted into a large complete graph. Modifying Wagner's proof, Dirac [lo] and Jung [17] proved that, for every natural number k there exists a natural number f , ( k ) such that every graph G of chromatic number at leastf,(k) contains a subdivision of the complete graph K, . These results have inspired several investigations on configurations, in particular, paths, cycles, and subdivision, in graphs or digraphs of large minimum degree, connectivity, or chromatic number. In the undirected case there is a richness of such results, while one very quickly encounters unsolved questions, counterexamples, and NP-complete problems in the directed case. Reference [39] gives a survey of these results and problems, and in the present paper we indicate more possible directions in this area. We also consider infinite graphs. In particular, we give a short proof of the recent result of Thomas [30] that the tree-width (to be defined later) of an infinite graph (of finite tree-width) is the maximum of the tree-widths of its finite subgraphs. Combined with results of Robertson and Seymour [ZS] this proves the conjecture of the author [38], as pointed out by Seese and Nesetril and Seymour (private communication) that every infinite graph of sufficiently large connectivity has a subgraph that is contractible to any prescribed finite planar graph.