Abstract
The tensor product $T(G_{1},G_{2},G_{3})$ of graphs $G_{1}$, $G_{2}$ and $G_{3}$ is defined by \[ VT(G_{1},G_{2},G_{3}) = V(G_{1}) \times V(G_{2}) \times V(G_{3}) \] and \[ ET(G_{1},G_{2},G_{3}) = \{ [(u_{1},u_{2},u_{3}), (v_{1},v_{2},v_{3})]: |\{ i: (u_{i},v_{i}) \in E(G_{i}) \}| \geq 2 \}. \] From this definition, it is easy to see that the preimage of the direct product of two independent sets of two factors under projections is an independent set of $T(G_{1}, G_2, G_3)$. So \[ \alpha T(G_{1},G_{2},G_{3}) \geq \max \{ \alpha(G_{1}) \alpha(G_{2}) |G_{3}|, \alpha(G_{1}) \alpha(G_{3}) |G_{2}|, \alpha(G_{2}) \alpha(G_{3}) |G_{1}| \}. \] In this paper, we prove that the equality holds if $G_{1}$ and $G_{2}$ are vertex-transitive graphs and $G_{3}$ is a circular graph, a Kneser graph, or a permutation graph. Furthermore, in this case, the structure of all maximum independent sets of $T(G_{1}$, $G_{2}$, $G_{3})$ is determined.
Funding Statement
Zhang was partially supported by the National Natural Science
Foundation of China (Nos. 11971439 and 12031018).
Acknowledgments
The authors thank Xuding Zhu for telling them the definition and problem on the tensor product of graphs.
Citation
Huiqun Mao. Huajun Zhang. "Independent Sets in Tensor Products of Three Vertex-transitive Graphs." Taiwanese J. Math. 25 (2) 207 - 222, April, 2021. https://doi.org/10.11650/tjm/210104
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