Hadwiger's conjecture [formula omitted] : Neighbour configurations of 6-vertices in contraction- critical graphs

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Publisher Summary The graphs considered in the chapter are simple (without loops or multiple edges). A vertex coloring is such that two neighbor vertices (joined by an edge) are of different colors; colors are designated by numbers 1, 2, 3…. A k -coloration of a graph G is a coloring using k different colors; when it is possible, G is said to be k -colorable. If G is k -colorable, but not ( k– 1)-colorable, it is said to be k -chromatic; k is the chromatic number of G. To contract a graph G consists in deleting the vertices and edges of a connected subgraph H of G, which is replaced with a new vertex h; the edges of G having one end in H are replaced with edges joining G–H with h, the other end remaining unchanged; multiple edges or loops occasionally resulting from this operation are eliminated. It is possible to carry out a contraction as a sequence of elementary contractions bearing upon one edge at once. Thus contraction is a transitive operation.