In Stanley's seminal 1995 paper on the chromatic symmetric function, he stated that there was no known graph that was not contractible to the claw and whose chromatic symmetric function was not $e$-positive, namely, not a positive linear combination of elementary symmetric functions. We resolve this by giving infinite families of graphs that are not contractible to the claw and whose chromatic symmetric functions are not $e$-positive. Moreover, one such family is additionally claw-free, thus establishing that the $e$-positivity of chromatic symmetric functions is in general not dependent on the existence of an induced claw or of a contraction to a claw.

中文翻译：

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解析斯坦利的 $e$-无爪收缩图的正性

在 Stanley 1995 年关于色对称函数的开创性论文中，他指出没有已知的图是不可收缩到爪子的，并且其色对称函数不是 $e$-positive，即不是初等对称函数的正线性组合. 我们通过给出不可收缩到爪子且其色对称函数不是 $e$-positive 的无限图族来解决这个问题。此外，还有一个这样的家族是无爪的，因此确定色对称函数的正性通常不依赖于诱导爪的存在或对爪的收缩。