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In graph theory, an acyclic coloring is a (proper) vertex coloring in which every 2-chromatic subgraph is acyclic. The acyclic chromatic number A(''G'') of a graph ''G'' is the least number of colors needed in any acyclic coloring of ''G''. Acyclic coloring is often associated with graphs embedded on non-plane surfaces. == Upper Bounds == A(''G'') ≤ 2 if and only if ''G'' is acyclic. Bounds on A(''G'') in terms of the maximum degree Δ(''G'') of ''G'' include the following: * A(''G'') ≤ 4 if Δ(''G'') = 3. * A(''G'') ≤ 5 if Δ(''G'') = 4. * A(''G'') ≤ 7 if Δ(''G'') = 5. * A(''G'') ≤ 12 if Δ(''G'') = 6. A milestone in the study of acyclic coloring is the following affirmative answer to a conjecture of Grünbaum: Theorem. :A(''G'') ≤ 5 if ''G'' is planar graph. introduced acyclic coloring and acyclic chromatic number, and conjectured the result in the above theorem. Borodin's proof involved several years of painstaking inspection of 450 reducible configurations. One consequence of this theorem is that every planar graph can be decomposed into an independent set and two induced forests. 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「Acyclic coloring」の詳細全文を読む スポンサード リンク
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