图的max- $k$ -割和色数的新界
New bounds for the max-$k$-cut and chromatic number of a graph
摘要：
我们考虑几个半定规划松弛的最大- $k$ -割问题，随着复杂性的增加。最弱的半定规划松弛的最优解有一个封闭形式的表达，其中包括最大的拉普拉斯特征值的图中考虑。这是当$k>2$适用于任何图时，最大$k$割的第一个已知的特征值界.利用这一界导出了图的色数的一个新的特征值界。对于正则图，新的色数界与著名的Hoffman界相同，但是，这两个界在一般情况下是不可比的。我们证明了几类图的max- $k$ -割的特征值界是紧的。我们研究了特定类别的图，如步行-正则图，强正则图，和从汉明关联计划的图形的界限。
英文摘要：
We consider several semidefinite programming relaxations for the max-$k$-cut problem, with increasing complexity. The optimal solution of the weakest presented semidefinite programming relaxation has a closed form expression that includes the largest Laplacian eigenvalue of the graph under consideration. This is the first known eigenvalue bound for the max-$k$-cut when $k>2$ that is applicable to any graph. This bound is exploited to derive a new eigenvalue bound on the chromatic number of a graph. For regular graphs, the new bound on the chromatic number is the same as the well-known Hoffman bound; however, the two bounds are incomparable in general. We prove that the eigenvalue bound for the max-$k$-cut is tight for several classes of graphs. We investigate the presented bounds for specific classes of graphs, such as walk-regular graphs, strongly regular graphs, and graphs from the Hamming association scheme.
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