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27
Hadwiger’s conjecture for K6free graphs
 COMBINATORICA
, 1993
"... In 1943, Hadwiger made the conjecture that every loopless graph not contractible to the complete graph on t + 1 vertices is tcolourable. When t ≤ 3 this is easy, and when t = 4, Wagner’s theorem of 1937 shows the conjecture to be equivalent to the fourcolour conjecture (the 4CC). However, when t ..."
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Cited by 34 (2 self)
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In 1943, Hadwiger made the conjecture that every loopless graph not contractible to the complete graph on t + 1 vertices is tcolourable. When t ≤ 3 this is easy, and when t = 4, Wagner’s theorem of 1937 shows the conjecture to be equivalent to the fourcolour 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 5colourable.
An Improved Linear Edge Bound for Graph Linkages
 EUROP. J. COMBINATORICS
, 2004
"... A graph is said to be klinked 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 2kconnected and ..."
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Cited by 28 (7 self)
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A graph is said to be klinked 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 2kconnected and G has at least 11kn edges, then G is klinked. We give a relatively simple inductive proof of the stronger statement that 8kn edges and 2kconnectivity suffice, and then with more effort improve the edge bound to 5kn.
On Sufficient Degree Conditions for a Graph to be klinked
, 2005
"... A graph is klinked 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,tipath. We consider degree conditions and connectivity conditions sufficient to force a graph to be klinked. Let D(n, k) be the minimum positive in ..."
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Cited by 12 (4 self)
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A graph is klinked 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,tipath. We consider degree conditions and connectivity conditions sufficient to force a graph to be klinked. Let D(n, k) be the minimum positive integer d such that every nvertex graph with minimum degree at least d is klinked and let R(n, k) be the minimum positive integer r such that every nvertex graph in which the sum of degrees of each pair of nonadjacent vertices is at least r is klinked. 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 klinked in terms of connectivity. Our bound allows us to modify the Thomas–Wollan proof slightly to show that every 2kconnected graph with average degree at least 12k is klinked.
Graph Connectivity after Path removal
"... Let G be a graph and u, v be two distinct vertices of G. A uv 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 ..."
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Cited by 7 (1 self)
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Let G be a graph and u, v be two distinct vertices of G. A uv 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 5connected graph, then for every pairof vertices u and v there exists a path P [u, v] such that G V (P [u, v]) is 2connected.
Large Topological Cliques in Graphs without a 4Cycle
 PROBAB. COMPUT
, 2002
"... Mader asked whether every C_4free 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... ..."
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Cited by 4 (4 self)
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Mader asked whether every C_4free 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...
On Minimum Degree Implying That a Graph Is HLinked
 SIAM J. DISCRETE MATH
, 2006
"... Given a fixed multigraph H, possibly containing loops, with V (H) ={h1,...,hm}, we say that a graph G is Hlinked if for every choice of m vertices v1,...,vm in G, there exists a subdivision of H in G such that vi is the branch vertex representing hi (for all i). This generalizes the concept of k ..."
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Cited by 3 (2 self)
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Given a fixed multigraph H, possibly containing loops, with V (H) ={h1,...,hm}, we say that a graph G is Hlinked if for every choice of m vertices v1,...,vm in G, there exists a subdivision of H in G such that vi is the branch vertex representing hi (for all i). This generalizes the concept of klinked graphs (as well as a number of other wellknown path or cycle properties). In this paper we determine a sharp lower bound on δ(G) (which depends upon H) such that each graph G on at least 10(V (H)  + E(H)) vertices satisfying this bound is Hlinked.
Subdivisions of K r+2 in graphs of average degree at least r + epsilon and large but constant girth
 Combin. Probab. Comput
"... 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 t ..."
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Cited by 3 (3 self)
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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.
Topological Minors in Graphs of Large Girth
 J. Combin. Theory B
, 1988
"... 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 ≥ 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 fa ..."
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Cited by 3 (2 self)
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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.