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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 36 (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.
Recent Excluded Minor Theorems for Graphs
 IN SURVEYS IN COMBINATORICS, 1999 267 201222. THE ELECTRONIC JOURNAL OF COMBINATORICS 8 (2001), #R34 8
, 1999
"... A graph is a minor of another if the first can be obtained from a subgraph of the second by contracting edges. An excluded minor theorem describes the structure of graphs with no minor isomorphic to a prescribed set of graphs. Splitter theorems are tools for proving excluded minor theorems. We disc ..."
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Cited by 9 (0 self)
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A graph is a minor of another if the first can be obtained from a subgraph of the second by contracting edges. An excluded minor theorem describes the structure of graphs with no minor isomorphic to a prescribed set of graphs. Splitter theorems are tools for proving excluded minor theorems. We discuss splitter theorems for internally 4connected graphs and for cyclically 5connected cubic graphs, the graph minor theorem of Robertson and Seymour, linkless embeddings of graphs in 3space, Hadwiger’s conjecture on tcolorability of graphs with no Kt+1 minor, Tutte’s edge 3coloring conjecture on edge 3colorability of 2connected cubic graphs with no Petersen minor, and Pfaffian orientations of bipartite graphs. The latter are related to the even directed circuit problem, a problem of Pólya about permanents, the 2colorability of hypergraphs, and signnonsingular matrices.
Recent Excluded Minor Theorems
 Surveys in Combinatorics, LMS Lecture Note Series
"... We discuss splitter theorems for internally 4connected graphs and for cyclically 5connected cubic graphs, the graph minor theorem, linkless embeddings, Hadwiger's conjecture, Tutte's edge 3coloring conjecture, and Pfaffian orientations of bipartite graphs. ..."
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Cited by 3 (1 self)
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We discuss splitter theorems for internally 4connected graphs and for cyclically 5connected cubic graphs, the graph minor theorem, linkless embeddings, Hadwiger's conjecture, Tutte's edge 3coloring conjecture, and Pfaffian orientations of bipartite graphs.
The Extremal Function for K9 Minors
, 2005
"... We prove that every (simple) graph on n ≥ 9 vertices and at least 7n − 27 edges either has a K9 minor, or is isomorphic to K2,2,2,3,3, or is isomorphic to a graph obtained from disjoint copies of K1,2,2,2,2,2 by identifying cliques of size six. The proof of one of our lemmas is computerassisted. 1 ..."
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Cited by 2 (0 self)
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We prove that every (simple) graph on n ≥ 9 vertices and at least 7n − 27 edges either has a K9 minor, or is isomorphic to K2,2,2,3,3, or is isomorphic to a graph obtained from disjoint copies of K1,2,2,2,2,2 by identifying cliques of size six. The proof of one of our lemmas is computerassisted. 1
K6 Minors in 6Connected Graphs . . .
, 2008
"... We prove that every sufficiently big 6connected graph of bounded treewidth either has a K6 minor, or has a vertex whose deletion makes the graph planar. This is a step toward proving that the same conclusion holds for all sufficiently big 6connected graphs. Jørgensen conjectured that it holds for ..."
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We prove that every sufficiently big 6connected graph of bounded treewidth either has a K6 minor, or has a vertex whose deletion makes the graph planar. This is a step toward proving that the same conclusion holds for all sufficiently big 6connected graphs. Jørgensen conjectured that it holds for all 6connected graphs.
K6 Minors in Large 6Connected Graphs
, 2012
"... Jørgensen conjectured that every 6connected graph G with no K6 minor has a vertex whose deletion makes the graph planar. We prove the conjecture for all sufficiently large graphs. ..."
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Jørgensen conjectured that every 6connected graph G with no K6 minor has a vertex whose deletion makes the graph planar. We prove the conjecture for all sufficiently large graphs.
and
, 2005
"... We prove that every sufficiently large 6connected graph of bounded treewidth either has a K6 minor, or has a vertex whose deletion makes the graph planar. This is a step toward proving that the same conclusion holds for all sufficiently large 6connected graphs. Jørgensen conjectured that it holds ..."
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We prove that every sufficiently large 6connected graph of bounded treewidth either has a K6 minor, or has a vertex whose deletion makes the graph planar. This is a step toward proving that the same conclusion holds for all sufficiently large 6connected graphs. Jørgensen conjectured that it holds for all 6connected graphs.
and
, 2005
"... Jørgensen conjectured that every 6connected graph G with no K6 minor has a vertex whose deletion makes the graph planar. We prove the conjecture for all sufficiently large graphs. ..."
Abstract
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Jørgensen conjectured that every 6connected graph G with no K6 minor has a vertex whose deletion makes the graph planar. We prove the conjecture for all sufficiently large graphs.
and
, 2005
"... We prove that every sufficiently big 6connected graph of bounded treewidth either has a K6 minor, or has a vertex whose deletion makes the graph planar. This is a step toward proving that the same conclusion holds for all sufficiently big 6connected graphs. Jørgensen conjectured that it holds for ..."
Abstract
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We prove that every sufficiently big 6connected graph of bounded treewidth either has a K6 minor, or has a vertex whose deletion makes the graph planar. This is a step toward proving that the same conclusion holds for all sufficiently big 6connected graphs. Jørgensen conjectured that it holds for all 6connected graphs.