Open Access
April, 2021 Independent Sets in Tensor Products of Three Vertex-transitive Graphs
Huiqun Mao, Huajun Zhang
Author Affiliations +
Taiwanese J. Math. 25(2): 207-222 (April, 2021). DOI: 10.11650/tjm/210104


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).


The authors thank Xuding Zhu for telling them the definition and problem on the tensor product of graphs.


Download Citation

Huiqun Mao. Huajun Zhang. "Independent Sets in Tensor Products of Three Vertex-transitive Graphs." Taiwanese J. Math. 25 (2) 207 - 222, April, 2021.


Received: 9 April 2019; Revised: 6 July 2020; Accepted: 11 October 2020; Published: April, 2021
First available in Project Euclid: 24 March 2021

Digital Object Identifier: 10.11650/tjm/210104

Primary: 05D05 , 06A07

Keywords: direct product , independence number , primitivity , tensor product , vertex-transitive graph

Rights: Copyright © 2021 The Mathematical Society of the Republic of China

Vol.25 • No. 2 • April, 2021
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