Robin Thomas

The Four-Colour Theorem

The four-colour theorem, that every loopless planar graph admits a vertex-colouring with at most four different colours, was proved in 1976 by Appel and Haken, using a computer. Here we give another proof, still using a computer, but simpler than Appel and Hakens in several respects.

Quickly excluding a planar graph

Abstract In an earlier paper, the first two authors proved that for any planar graph H , every graph with no minor isomorphic to H has bounded tree width; but the bound given there was enormous. Here we prove a much better bound. We also improve the best known bound on the tree-width of planar graphs with no minor isomorphic to a g × g grid.

Call routing and the ratcatcher

Suppose we expect there to bep(ab) phone calls between locationsa andb, all choices ofa, b from some setL of locations. We wish to design a network to optimally handle these calls. More precisely, a “routing tree” is a treeT with set of leavesL, in which every other vertex has valency 3. It has “congestion” <k if for every edgee ofT, there are fewer thank calls which will be routed alonge, that is, between locationa, b in different components ofT/e. Deciding if there is a routing tree with congestion <k is NP-hard, but if the pairsab, withp(ab)>0 form the edges of a planar graphG, there is an efficient, strongly polynomial algorithm.This is because the problem is equivalent to deciding if a ratcatcher can corner a rat loose in the walls of a house with floor planG, wherep(ab) is a thickness of the wallab. The ratcatcher carries a noisemaker of powerk, and the rat will not move through any wall in which the noise level is too high (determined by the total thickness of the intervening walls between this one and the noisemaker).It follows that branch-width is polynomially computable for planar graphs—that too is NP-hard for general graphs.

Directed Tree-Width

We generalize the concept of tree-width to directed graphs and prove that every directed graph with no “haven” of large order has small tree-width. Conversely, a digraph with a large haven has large tree-width. We also show that the Hamilton cycle problem and other NP-hard problems can be solved in polynomial time when restricted to digraphs of bounded tree-width.

Hadwiger's conjecture for K 6 -free graphs.

In 1943, Hadwiger made the conjecture that every loopless graph not contractible to the complete graph ont+1 vertices ist-colourable. Whent≤3 this is easy, and whent=4, Wagners theorem of 1937 shows the conjecture to be equivalent to the four-colour conjecture (the 4CC). However, whent≥5 it has remained open. Here we show that whent=5 it is also equivalent to the 4CC. More precisely, we show (without assuming the 4CC) that every minimal counterexample to Hadwigers conjecture whent=5 is “apex”, that is, it consists of a planar graph with one additional vertex. Consequently, the 4CC implies Hadwigers conjecture whent=5, because it implies that apex graphs are 5-colourable.

A separator theorem for graphs with an excluded minor and its applications

LetG be an n-vertex graph with nonnegative weights whose sum is 1 assigned to its vertices, and with no minor isomorphic to a given h-vertex graph H. We prove that there is a set X of no more than hn vertices of G whose deletion creates a graph in which the total weight of every connected component is at most 1/2. This extends significantly a well-known theorem of Lipton and Tarjan for planar graphs. We exhibit an algorithm which finds, given an n-vertex graph G with weights as above and an h-vertex graph H, either such a set X or a minor of G isomorphic to H. The algorithm runs in time O(hnm), where m is the number of edges of G plus the number of its vertices. Our results supply extensions of the many known applications of the Lipton-Tarjan separator theorem from the class of planar graphs (or that of graphs with bounded genus) to any class of graphs with an excluded minor. For example, it follows that for any fixed graph H , given a graph G with n vertices and with no H-minor one can approximate the size of the maximum independent set of G up to a relative error of 1/ √ log n in polynomial time, find that size exactly and find the chromatic number of G in time 2 √ n) and solve any sparse system of n linear equations in n unknowns whose sparsity structure 0 corresponds to G in time O(n). We also describe a combinatorial application of our result which relates the tree-width of a graph to the maximum size of a Kh-minor in it.

On the complexity of finding iso- and other morphisms for partial k -trees

Abstract The problems to decide whether H ⩽ G for input graphs H , G where ⩽ is ‘isomorphic to a subgraph’, ‘isomorphic to an induced subgraphs’, ‘isomorphic to a subdivision’, ‘isomorphic to a contraction’ or their combination, are NP-complete. We discuss the complexity of these problems when G is restricted to be a partial k -tree (in other terminology: to have tree-width ⩽ k , to be k -decomposable, to have dimension ⩽ k ). Under this restriction the problems are still NP-complete in general, but there are polynomial algorithms under some natural restrictions imposed on H , for example when H has bounded degrees. We also give a polynomial time algorithm for the n disjoint connecting paths problem restricted to partial k -trees (with n part of input).

4-Connected Projective-Planar Graphs Are Hamiltonian

We prove the result stated in the title (conjectured by Grunbaum) and a conjecture of Plummer that every graph which can be obtained from a 4-connected planar graph by deleting two vertices is Hamiltonian. The proofs are constructive and give rise to polynomial-time algorithms.

A new proof of the four-colour theorem

The four-colour theorem, that every loopless planar graph admits a vertex-colouring with at most four different colours, was proved in 1976 by Appel and Haken, using a computer. Here we announce another proof, still using a computer, but simpler than Appel and Haken’s in several respects.

An improved linear edge bound for graph linkages

A graph is said to be k-linked if it has at least 2k vertices and for every sequence S1,...,Sk, t1,...,tk of distinct vertices there exist disjoint paths P1,...,Pk such that the ends of Pi are si and ti. Bollobas and Thomason showed that if a simple graph G on n vertices is 2k-connected and G has at least 11kn edges, then G is k-linked. We give a relatively simple inductive proof of the stronger statement that 8kn edges and 2k-connectivity suffice, and then with more effort improve the edge bound to 5kn.