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.

A graph is said to be k-linked if it has at least 2k vertices and for every sequence s1,...,s k,t 1,...,t k of distinct vertices there exist disjoint paths P1,...,P k such that the ends of P i are s i and t i . 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.

A graph is k-linked if for every list of 2k vertices {s1,...,sk,t1,...,tk}, there exist internally disjoint paths P1,...,Pk such that each Pi is an si,ti-path. We consider degree conditions and connectivity conditions sufficient to force a graph to be k-linked. Let D(n, k) be the minimum positive integer d such that every n-vertex graph with minimum degree at least d is k-linked and let R(n, k) be the minimum positive integer r such that every n-vertex graph in which the sum of degrees of each pair of non-adjacent vertices is at least r is k-linked. The main result of the paper is finding the exact values of D(n, k) andR(n, k) for every n and k. Thomas and Wollan [14] used the bound D(n, k) � (n +3k)/2 − 2 to give sufficient conditions for a graph to be k-linked in terms of connectivity. Our bound allows us to modify the Thomas–Wollan proof slightly to show that every 2k-connected graph with average degree at least 12k is k-linked.

DOI 10.1002/jgt.20115

Let G be a graph and u, v be two distinct vertices of G. A u-v path P is called nonseparatingif G- V (P) is connected. The purpose of this paper is to study the number of nonseparating u- v path for two arbitrary vertices u and v of a given graph. For a positive integer k, we willshow that there is a minimum integer ff(k) so that if G is an ff(k)-connected graph and u and v are two arbitrary vertices in G, then there exist k vertex disjoint paths P1[u, v], P2[u, v],..., Pk[u, v] such that G- V (Pi[u, v]) is connected for every i (i = 1, 2,..., k). In fact, we will provethat ff(k) < = 22k + 2. It is known that ff(1) = 3. A result of Tutte showed that ff(2) = 3. Weshow that ff(3) = 6. In addition, we prove that if G is a 5-connected graph, then for every pairof vertices u and v there exists a path P [u, v] such that G- V (P [u, v]) is 2-connected.

Mader asked whether every C_4-free graph G contains a subdivision of a complete graph whose order is at least linear in the average degree of G. We show that there is a subdivision...

We show that for every # > 0 there exists an r 0 = r 0 (#) such that for all integers r r 0 every graph of average degree at least r + # and girth at least 1000 contains a subdivision of K r+2 . Combined with a result of Mader this implies that for every # > 0 there exists an f(#) such that for all r 2 every graph of average degree at least r + # and girth at least f(#) contains a subdivision of K r+2 . We also prove a more general result concerning subdivisions of arbitrary graphs.

We prove that every graph of minimum degree at least r and girth at least 186 contains a subdivision of K_r+1 and that for r &ge; 435 a girth of at least 15 suces. This implies that the conjecture of Hajós that every graph of chromatic number at least r contains a subdivision of K_r (which is false in general) is true for graphs of girth at least 186 (or 15 if r &ge; 436). More generally, we show that for every graph H of maximum degree &Delta;(H) &ge; 2, every graph G of minimum degree at least max{&Delta;(H), 3} and girth at least 166 ... contains a subdivision of H . This bound on the girth of G is best possible up to the value of the constant and improves a result of Mader, who gave a bound linear in |H|.

Bollobás and Thomason showed that every 22k-connected graph is k-linked. Their result used a dense graph minor. In this paper we investigate the ties between small graph minors and linkages. In particular, we show that a 6-connected graph with a K − 9 minor is 3-linked. Further, we show that a 7-connected graph with a K − 9 minor is (2, 5)-linked. Finally, we show that a graph of order n and size at least 7n − 29 contains a K −− 9 minor. 1

A graph is k-linked if for every set of 2k distinct vertices {s1,..., sk, t1,..., tk} there exist disjoint paths P1,..., Pk such that the endpoints of Pi are si and ti. We prove every 6-connected graph on n vertices with 5n − 14 edges is 3-linked. This is optimal, in that there exist 6-connected graphs on n vertices with 5n − 15 edges that are not 3-linked for arbitrarily large values of n.