We consider the dynamic graph coloring problem restricted to the class of interval graphs in the incremental and fully dynamic setting. The input consists of a sequence of intervals that are to be either colored, or deleted, if previously colored. For the incremental setting, we consider the well studied optimal online algorithm (KT-algorithm) for interval coloring due to Kierstead and Trotter [1]. We present the following results on the dynamic interval coloring problem.

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Any direct implementation of the KT-algorithm requires $\mathrm{\Omega }({\mathrm{\Delta }}^2)$ time per interval in the worst case.

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There exists an incremental algorithm which supports insertion of an interval in amortized $O(\log n+\mathrm{\Delta })$ time per update and maintains a proper coloring using at most $3\omega -2$ colors.

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There exists a fully dynamic algorithm which supports insertion of an interval in $O(\log n+\mathrm{\Delta }\log \omega )$ update time and deletion of an interval in $O({\mathrm{\Delta }}^2\log n)$ update time in the worst case and maintains a proper coloring using at most $3\omega -2$ colors.

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Any algorithm that computes the induced subgraph among the neighbors of a given vertex requires at least quadratic time unless the OMv conjecture [2] is false.

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If the matrix and the vectors in the online sequence have the consecutive ones property, then the OMv conjecture [2] is false.

我们认为动态图着色问题仅限于增量和完全动态设置中的间隔图类。输入由一系列间隔组成，这些间隔将被着色或删除（如果先前已着色）。对于增量设置，由于Kierstead和Trotter [1]，我们考虑对间隔着色进行了充分研究的最佳在线算法（KT-algorithm）。我们提出以下有关动态间隔着色问题的结果。

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KT算法的任何直接实施都需要$\mathrm{\Omega }（{\mathrm{\Delta }}^2）$ 最坏情况下每个时间间隔的时间。

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存在一种增量算法，该算法支持在摊销中插入间隔 $Ø（\mathrm{日志}ñ+\mathrm{\Delta }）$ 每次更新的时间，并且最多使用一次来保持正确的着色 $3\omega -2$ 颜色。

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存在一种全动态算法，该算法支持在 $Ø（\mathrm{日志}ñ+\mathrm{\Delta }\mathrm{日志}\omega ）$ 更新时间和删除间隔 $Ø（{\mathrm{\Delta }}^2\mathrm{日志}ñ）$ 在最坏的情况下更新时间，并最多使用一次来保持正确的着色 $3\omega -2$ 颜色。

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除非OMv猜想[2]为假，否则任何计算给定顶点邻居之间的诱导子图的算法都至少需要二次时间。

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如果在线序列中的矩阵和向量具有连续的1属性，则OMv猜想[2]为假。