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arXiv · 1605.01978

An inertial lower bound for the chromatic number of a graph

Clive Elphick·Pawel Wocjan
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Abstract

Let $\chi(G$) and $\chi_f(G)$ denote the chromatic and fractional chromatic numbers of a graph $G$, and let $(n^+ , n^0 , n^-)$ denote the inertia of $G$. We prove that: \[ 1 + \max\left(\frac{n^+}{n^-} , \frac{n^-}{n^+}\right) \le \chi(G) \mbox{ and conjecture that } 1 + \max\left(\frac{n^+}{n^-} , \frac{n^-}{n^+}\right) \le \chi_f(G) \] We investigate extremal graphs for these bounds and demonstrate that this inertial bound is not a lower bound for the vector chromatic number. We conclude with a discussion of asymmetry between $n^+$ and $n^-$, including some Nordhaus-Gaddum bounds for inertia.

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Clive Elphick, Pawel Wocjan. 2016-05-06. An inertial lower bound for the chromatic number of a graph. https://arxiv.org/abs/1605.01978

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