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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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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.
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"... If F is a finite family of sets, then the intersection graph r(F) is the graph with vertexset F and edges the unordered pairs C, D of distinct elements of F such that C n D # 0. It is easy to see [6, p. 19] that every graph G is isomorphic to some intersection graph T(F). Some interesting classes o ..."
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If F is a finite family of sets, then the intersection graph r(F) is the graph with vertexset F and edges the unordered pairs C, D of distinct elements of F such that C n D # 0. It is easy to see [6, p. 19] that every graph G is isomorphic to some intersection graph T(F). Some interesting classes of graphs have arisen by letting F range over families of balls in