The construction of disjoint paths in a network is a basic issue in combinatorial optimization: given a network, and specified pairs of nodes in it, we are interested in finding disjoint paths between as many of these pairs as possible. This leads to a variety of classical NP-complete problems for which very little is known from the point of view of approximation algorithms. It has recently been brought into focus in work on problems such as VLSI layout and routing in high-speed networks; in these settings, the current lack of understanding of the disjoint paths problem is often an obstacle to the design of practical heuristics.

this article the structure of the models of decidable (weak) monadic theories of planar graphs is investigated. It is shown that if the (weak) monadic theory of a class K of planar graphs is decidable, then the tree-width in the sense of Robertson and Seymour (1984) of the elements of K is universally bounded and there is a class T of trees such that the (weak) monadic theory of K is interpretable in the (weak) monadic theory of T

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

Finite graph homology may seem trivial, but for infinite graphs things become interesting. We present a new approach that builds the cycle space of a graph not on its finite cycles but on its topological circles, the homeomorphic images of the unit circle in the space formed by the graph together with its ends. Our approach

A recently posed question of Häggkvist and Scott’s asked whether or not there exists a constant c such that if G is a graph of minimum degree ck then G contains cycles of k consecutive even lengths. In this paper we answer the question by proving that for k ≥ 2, a bipartite graph of average degree at least 4k and girth g contains cycles of (g/2 − 1)k consecutive even lengths. We also obtain a short proof of the theorem of Bondy and Simonovits, that a graph of order n and size at least 8(k −1)n 1+1/k has a cycle of length 2k. Erdős and Burr [4] conjectured that for every odd number k, there is a constant ck such that for every natural number m, every graph of average degree at least ck contains a cycle of length m modulo k. Erdős and Burr [4] settled their conjecture in the case m = 2 and Robertson (see [4]) settled the case m = 0. The full conjecture was resolved by Bollobás [1], who proved the conjecture with ck = 2[(k + 1) k − 1]/k. In this paper, we show that ck = 8k will do. Thomassen [11] later showed cycles of all even lengths modulo k are obtained under the hypothesis that the average degree is at least 4k(k + 1), without requiring k to be odd. Thomassen [10] also proved that if G is a graph of minimum degree at least three and girth at least

this paper is to reduce the upper bound for the required girth to the correct order of magnitude: Theorem 1. For any inte k,e ve graph G of girth g(G) > 6 log k +3and #(G) # 3 has a minor H with #(H) >k. The best lower bound we have found is 8 3 log c, but we note that existing conjectures about cubic graphs of large girth would raise this to about 4 log . Since an average degree of at least cr # log r forces a K r minor [ 5, 8 ], Theorem 1 has the following consequence: Corollary 2.The ee a constant c # R such thate ea graph G of girth g(G) # 6 log r<F1

We show that for every odd integer g 5 there exists a constant c such that every graph of minimum degree r and girth at least g contains a minor of minimum degree at least cr . This is best possible up to the value of the constant c for g = 5; 7 and 11. More generally, a wellknown conjecture about the minimal order of graphs of given minimum degree and large girth would imply that our result gives the correct order of magnitude for all odd values of g. The case g = 5 of our result implies Hadwiger's conjecture for C 4 -free graphs of suciently large chromatic number.

A simple graph H is a cover of a graph G if there exists a mapping ϕ from H onto G such that ϕ maps the neighbors of every vertex v in H bijectively to the neighbors of ϕ(v) in G. Negami conjectured in 1986 that a connected graph has a finite planar cover if and only if it embeds in the projective plane. The conjecture is still open. It follows from the results of Archdeacon, Fellows, Negami, and the first author that the conjecture holds as long as the graph K1,2,2,2 has no finite planar cover. However, those results seem to say little about counterexamples if the conjecture was not true. We show that there are, up to obvious constructions, at most 16 possible counterexamples to Negami’s conjecture. Moreover, we exhibit a finite list of sets of graphs such that the set of excluded minors for the property of having finite planar cover is one of the sets in our list.

We give axiomatic foundations for non-finitary infinite matroids with duality, in terms of independent sets, bases, circuits, closure and rank. This completes the solution to a problem of Rado of 1966.

Let G be a graph, {a, b, c} ⊆ V (G), and {a ′ , b ′ , c ′ } ⊆ V (G) such that {a, b, c} ̸= {a ′ , b ′ , c ′}. We say that (G, {a, c}, {a ′ , c ′}, (b, b ′)) is an obstruction if, for any three vertex disjoint paths from {a, b, c} to {a ′ , b ′ , c ′ } in G, one path is from b to b ′. Robertson and Seymour asked the problem of characterizing all obstructions. In this paper, we present a list of “basic ” obstructions and show how to produce other obstructions from these basic ones. We also prove results about disjoint paths in graphs. Results in this paper will be used in subsequent papers to characterize all obstructions. Partially supported by NSF grant DMS-9970527 and NSA grant MSPF-02G-074