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Graphs and Combinatorics

, Volume 7, Issue 1, pp 65–87 | Cite as

Symmetric edge-decompositions of hypercubes

  • Mark Ramras
Article

Abstract

We call a set of edgesE of the n-cubeQ n a fundamental set for Q n if for some subgroupG of the automorphism group ofQ n , theG-translates ofE partition the edge set ofQ n .Q n possesses an abundance of fundamental sets. For example, a corollary of one of our main results is that if |E| =n and the subgraph induced byE is connected, then if no three edges ofE are mutually parallel,E is a fundamental set forQ n . The subgroupG is constructed explicitly. A connected graph onn edges can be embedded intoQ n so that the image of its edges forms such a fundamental set if and only if each of its edges belongs to at most one cycle.

We also establish a necessary condition forE to be a fundamental set. This involves a number-theoretic condition on the integersa j (E), where for 1 ≤jn, a j (E) is the number of edges ofE in thej th direction (i.e. parallel to thej th coordinate axis).

Keywords

Automorphism Group Coordinate Axis Connected Graph Edge Form Condition forE 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag 1991

Authors and Affiliations

  • Mark Ramras
    • 1
  1. 1.Department of MathematicsNortheastern UniversityBostonUSA

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