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:''Not to be confused with the Critical path method, a concept in project management.'' In general the notion of ''criticality'' can refer to any measure. But in graph theory, when the term is used without any qualification, it almost always refers to the chromatic number of a graph. Critical graphs are interesting because they are the ''minimal'' members in terms of chromatic number, which is a very important measure in graph theory. More precise definitions follow. A vertex or an edge is a critical element of a graph ''G'' if its deletion would decrease the chromatic number of ''G''. Obviously such decrement can be no more than 1 in a graph. A critical graph is a graph in which every vertex or edge is a critical element. A ''k''-critical graph is a critical graph with chromatic number ''k''; a graph ''G'' with chromatic number ''k'' is ''k''-vertex-critical if each of its vertices is a critical element. Some properties of a ''k''-critical graph ''G'' with ''n'' vertices and ''m'' edges: * ''G'' has only one component. * ''G'' is finite (this is the de Bruijn–Erdős theorem of ). * δ(''G'') ≥ ''k'' − 1, that is, every vertex is adjacent to at least ''k'' − 1 others. More strongly, ''G'' is (''k'' − 1)-edge-connected. See * If ''G'' is (''k'' − 1)-regular, meaning every vertex is adjacent to exactly ''k'' − 1 others, then ''G'' is either ''Kk'' or an odd cycle . This is Brooks' theorem; see ). * 2 ''m'' ≥ (''k'' − 1) ''n'' + ''k'' − 3 . * 2 ''m'' ≥ (''k'' − 1) ''n'' + (− 3)/(''k''2 − 3) ) ''n'' . * Either ''G'' may be decomposed into two smaller critical graphs, with an edge between every pair of vertices that includes one vertex from each of the two subgraphs, or ''G'' has at least 2''k'' - 1 vertices . More strongly, either ''G'' has a decomposition of this type, or for every vertex ''v'' of ''G'' there is a ''k''-coloring in which ''v'' is the only vertex of its color and every other color class has at least two vertices . It is fairly easy to see that a graph G is vertex-critical if and only if for every vertex ''v'', there is an optimal proper coloring in which ''v'' is a singleton color class. As showed, every ''k''-critical graph may be formed from a complete graph ''K''''k'' by combining the Hajós construction with an operation of identifying two nonadjacent vertices. The graphs formed in this way always require ''k'' colors in any proper coloring. A double-critical graph is a connected graph in which the deletion of any pair of adjacent vertices decreases the chromatic number by two. One open problem is to determine whether ''K''''k'' is the only double-critical ''k''-chromatic graph . == See also == * Factor-critical graph 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「Critical graph」の詳細全文を読む スポンサード リンク
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