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The Thickness of the Cartesian Product of Two Graphs

Published online by Cambridge University Press:  20 November 2018

Yichao Chen  and
Xuluo Yin
Yichao Chen
Affiliation:
Department of Mathematics, Hunan University, 410082 Changsha, China e-mail: ycchen@hnu.edu.cn e-mail: xuluoyin@hnu.edu.cn
Xuluo Yin
Affiliation:
Department of Mathematics, Hunan University, 410082 Changsha, China e-mail: ycchen@hnu.edu.cn e-mail: xuluoyin@hnu.edu.cn
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Abstract

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The thickness of a graph $G$ is the minimum number of planar subgraphs whose union is $G$. A $t$-minimal graph is a graph of thickness $t$ that contains no proper subgraph of thickness $t$. In this paper, upper and lower bounds are obtained for the thickness, $t\left( G\,\square \,H \right)$, of the Cartesian product of two graphs $G$ and $H$, in terms of the thickness $t\left( G \right)$ and $t\left( H \right)$. Furthermore, the thickness of the Cartesian product of two planar graphs and of a $t$-minimal graph and a planar graph are determined. By using a new planar decomposition of the complete bipartite graph ${{K}_{4k,\,4k}}$, the thickness of the Cartesian product of two complete bipartite graphs ${{K}_{n,n}}$ and ${{K}_{n,n}}$ is also given for $n\,\ne \,4k\,+\,1$.

Keywords

05C10planar graphthicknessCartesian productt-minimal graphcomplete bipartite graph
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 59 , Issue 4 , 01 December 2016 , pp. 705 - 720
Copyright
Copyright © Canadian Mathematical Society 2016

References

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