Abstract
The thickness of a graph is the minimum number of planar spanning subgraphs into which the graph can be decomposed. It is known for relatively few classes of graphs, compared to other topological invariants, e.g., genus and crossing number. For the complete bipartite graphs, Beineke et al. (Proc Camb Philos Soc 60:1–5, 1964) gave the answer for most graphs in this family in 1964. In this paper, we derive formulas and bounds for the thickness of some complete k-partite graphs. And some properties for the thickness for the join of two graphs are also obtained.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Alekseev VB, Gončhakov VS (1976) The thickness of an arbitrary complete graph. Math Sbornik 30(2):187–202
Beineke LW, Harary F (1965) The thickness of the complete graph. Can J Math 17:850–859
Beineke LW, Harary F, Moon JW (1964) On the thickness of the complete bipartite graph. Proc Camb Philos Soc 60:1–5
Mansfield A (1983) Determining the thickness of graphs is NP-hard. Math Proc Camb Philos Soc 93(9):9–23
Mutzel P, Odenthal T, Scharbrodt M (1998) The thickness of graphs: a survey. Graph Combin 14:59–73
Poranen T (2005) A simulated annealing algorithm for determining the thickness of a graph. Inf Sci 172:155–172
Tutte WT (1963) The thickness of a graph. Indag Math 25:567–577
Vasak JM (1976) The thickness of the complete graph. Not Am Math Soc 23:A-479
Yang Y (2014) A note on the thickness of \(K_{l, m, n}\). Ars Comb 117:349–351
Acknowledgments
The work of the first author was supported by NNSFC under Grants No. 11471106.
The work of the second author was supported by NNSFC under Grants No. 11401430.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Chen, Y., Yang, Y. The thickness of the complete multipartite graphs and the join of graphs. J Comb Optim 34, 194–202 (2017). https://doi.org/10.1007/s10878-016-0057-1
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10878-016-0057-1