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On Homeomorphic Embeddings of Km,n in the Cube

Published online by Cambridge University Press:  20 November 2018

Jehuda Hartman
Jehuda Hartman*
Affiliation:
The University of Alberta, Edmonton, Alberta
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Homeomorphic embeddings of Knin the m-cube were investigated in [6]. In particular, it was proved that any homeomorph of Kn+1embedded in the m-cube has at least n2edges. Furthermore, homeomorphic embeddings of Kn+1having exactly n2edges are unique up to isomorphism. In this paper a similar problem for the complete bipartite graph is considered.

We adopt the notation and terminology of [5].

All graphs considered are without loops and multiple edges.

Let x = uv be an edge of a graph G; x will be called subdivided if it is replaced by a vertex ω and by edges uω and ωv. A graph G’ is called a subdivision of G if it is obtained from G by a subdivision of an edge of G. A refinement Ĝ of G, is a graph isomorphic to a graph obtained from G by a finite sequence of subdivisions.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 32 , Issue 3 , 01 June 1980 , pp. 644 - 652
Copyright
Copyright © Canadian Mathematical Society 1980

References

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