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Fractional chromatic numbers of tensor products of three graphs

Authors:	Jimeng Xiao Huajun Zhang Shenggui Zhang

Abstract:	The tensor product $(G_{1}, G_{2}, G_{3})$ of graphs $G_{1}$ , $G_{2}$ and $G_{3}$ is defined by $V (G_{1}, G_{2}, G_{3}) = V (G_{1}) \times V (G_{2}) \times V (G_{3})$ and $E (G_{1}, G_{2}, G_{3}) = \{((u_{1}, u_{2}, u_{3}), (v_{1}, v_{2}, v_{3})) : \| {i : (u_{i}, v_{i}) \in E (G_{i})} \| \geq 2\} .$ Let $χ_{f} (G)$ be the fractional chromatic number of a graph $G$ . In this paper, we prove that if one of the three graphs $G_{1}$ , $G_{2}$ and $G_{3}$ is a circular clique, $χ_{f} (G_{1}, G_{2}, G_{3}) = min {χ_{f} (G_{1}) χ_{f} (G_{2}), χ_{f} (G_{1}) χ_{f} (G_{3}), χ_{f} (G_{2}) χ_{f} (G_{3})} .$

Keywords:	Corresponding author Direct product Tensor product Fractional chromatic number Fractional clique number Circular clique
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