Carsten Thomassen

Every Planar Graph Is 5-Choosable

We prove the statement of the title, which was conjectured in 1975 by V. G. Vizing and, independently, in 1979 by P. Erdos, A. L. Rubin, and H Taylor.

The graph genus problem is NP-complete

Abstract It is NP-compete to tell, given a graph G and a natural number k , whether G has genus k or less.

Cycles in digraphs– a survey

The main subjects of this survey paper are Hamitonian cycles, cycles of prescirbed lengths, cycles in tournaments, and partitions, packings, and coverings by cycles. Several unsolved problems and a bibiligraphy are included.

Planarity and Duality of Finite and Infinite Graphs

Abstract We present a short proof of the following theorems simultaneously: Kuratowskis theorem, Farys theorem, and the theorem of Tutte that every 3-connected planar graph has a convex representation. We stress the importance of Kuratowskis theorem by showing how it implies a result of Tutte on planar representations with prescribed vertices on the same facial cycle as well as the planarity criteria of Whitney, MacLane, Tutte, and Fournier (in the case of Whitneys theorem and MacLanes theorem this has already been done by Tutte). In connection with Tuttes planarity criterion in terms of non-separating cycles we give a short proof of the result of Tutte that the induced non-separating cycles in a 3-connected graph generate the cycle space. We consider each of the above-mentioned planarity criteria for infinite graphs. Specifically, we prove that Tuttes condition in terms of overlap graphs is equivalent to Kuratowskis condition, we characterize completely the infinite graphs satisfying MacLanes condition and we prove that the 3-connected locally finite ones have convex representations. We investigate when an infinite graph has a dual graph and we settle this problem completely in the locally finite case. We show by examples that Tuttes criterion involving non-separating cycles has no immediate extension to infinite graphs, but we present some analogues of that criterion for special classes of infinite graphs.

Highly Connected Sets and the Excluded Grid Theorem

We present a short proof of the excluded grid theorem of Robertson and Seymour, the fact that a graph has no large grid minor if and only if it has small tree-width. We further propose a very simple obstruction to small tree-width inspired by that proof, showing that a graph has small tree-width if and only if it contains no large highly connected set of vertices.

Reflections on graph theory

At the occasion of the 250th anniversary of graph theory, we recall some of the basic results and unsolved problems, some of the attractive and surprising methods and results, and some possible future directions in graph theory.

A theorem on paths in planar graphs

We prove a theorem on paths with prescribed ends in a planar graph which extends Tuttes theorem on cycles in planar graphs [9] and implies the conjecture of Plummer [5] asserting that every 4-connected planar graph is Hamiltonian-connected.

3-list-coloring planar graphs of girth 5

Abstract We prove that every planar graph of girth at least 5 is 3-choosable. It is even possible to precolor any 5-cycle in the graph. This extension implies Grotzsch′s theorem that every planar graph of girth at least 4 is 3-colorable.

Interval representations of planar graphs

Abstract We prove that every planar graph is the intersection graph of a collection of three-dimensional boxes, with intersections occuring only in the boundaries of the boxes. Furthermore, we characterize the graphs that have such representations (called strict representations) in the plane. These are precisely the proper subgraphs of 4-connected planar triangulations, which we characterize by forbidden sub-graphs. Finally, we strengthen a result of E. R. Scheinerman (“Intersection Classes and Multiple Intersection Parameters”, Ph. D. thesis, Princeton Univ., 1984) to show that every planar graph has a strict representation using at most two rectangles per vertex.

Embeddings of graphs with no short noncontractible cycles

Abstract We investigate embeddings of graphs on orientable 2-dimensional surfaces such that all face boundaries have fewer edges than every noncontractible cycle. We show that such embeddings are always minimum genus embeddings and that they share many properties with planar embeddings. For example, if the graph is 3-connected, then the embedding is unique. We use this to obtain a polynomially bounded algorithm for describing a minimum genus embedding with no short non-contractible cycles if such an embedding of the graph exists. We refine some of these results for triangulations.