量子色数的估计
Estimating quantum chromatic numbers
摘要：
进一步发展了第一和第四作者提出的图的量子色数的新形式。证明了图的可交换量子色数的计算问题可以用SDP算法求解，并描述了收敛于图的可交换量子色数的一系列变体，引入了图的迹秩，并证明了它是乘法的.我们以统一的方式描述了迹秩、投射秩和分数色数，阐明了它们分别与可交换量子色数、量子色数和经典色数的联系。最后，我们提出了一个新的SDP算法，产生一个参数大于Lovász数，但仍是一个下界的图的tracial秩。我们确定了奇循环迹秩的精确值。
英文摘要：
We develop further the new versions of quantum chromatic numbers of graphs introduced by the first and fourth authors. We prove that the problem of computation of the commuting quantum chromatic number of a graph is solvable by an SDP algorithm and describe an hierarchy of variants of the commuting quantum chromatic number which converge to it. We introduce the tracial rank of a graph, a parameter that gives a lower bound for the commuting quantum chromatic number and parallels the projective rank, and prove that it is multiplicative. We describe the tracial rank, the projective rank and the fractional chromatic numbers in a unified manner that clarifies their connection with the commuting quantum chromatic number, the quantum chromatic number and the classical chromatic number, respectively. Finally, we present a new SDP algorithm that yields a parameter larger than the Lovász number and is yet a lower bound for the tracial rank of the graph. We determine the precise value of the tracial rank of an odd cycle.
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