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In graph theory, the Hadwiger conjecture (or Hadwiger's conjecture) states that, if all proper colorings of an undirected graph ''G'' use ''k'' or more colors, then one can find ''k'' disjoint connected subgraphs of ''G'' such that each subgraph is connected by an edge to each other subgraph. Contracting the edges within each of these subgraphs so that each subgraph collapses to a single vertex produces a complete graph ''Kk'' on ''k'' vertices as a minor of ''G''. This conjecture, a far-reaching generalization of the four-color problem, was made by Hugo Hadwiger in 1943 and is still unsolved. call it “one of the deepest unsolved problems in graph theory.”〔 ==Equivalent forms== An equivalent form of the Hadwiger conjecture (the contrapositive of the form stated above) is that, if there is no sequence of edge contractions (each merging the two endpoints of some edge into a single supervertex) that brings a graph ''G'' to the complete graph ''Kk'', then ''G'' must have a vertex coloring with ''k'' − 1 colors. Note that, in a minimal ''k''-coloring of any graph ''G'', contracting each color class of the coloring to a single vertex will produce a complete graph ''Kk''. However, this contraction process does not produce a minor of ''G'' because there is (by definition) no edge between any two vertices in the same color class, thus the contraction is not an edge contraction (which is required for minors). Hadwiger's conjecture states that there exists a different way of properly edge contracting sets of vertices to single vertices, producing a complete graph ''Kk'', in such a way that all the contracted sets are connected. If ''Fk'' denotes the family of graphs having the property that all minors of graphs in ''Fk'' can be (''k'' − 1)-colored, then it follows from the Robertson–Seymour theorem that ''Fk'' can be characterized by a finite set of forbidden minors. Hadwiger's conjecture is that this set consists of a single forbidden minor, ''Kk''. The Hadwiger number ''h''(''G'') of a graph ''G'' is the size ''k'' of the largest complete graph ''Kk'' that is a minor of ''G'' (or equivalently can be obtained by contracting edges of ''G''). It is also known as the contraction clique number of ''G''.〔.〕 The Hadwiger conjecture can be stated in the simple algebraic form ''χ''(''G'') ≤ ''h''(''G'') where ''χ''(''G'') denotes the chromatic number of ''G''. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Hadwiger conjecture (graph theory)」の詳細全文を読む スポンサード リンク
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