Symmetric edge-decompositions of hypercubes
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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 ≤j ≤n, a j (E) is the number of edges ofE in thej th direction (i.e. parallel to thej th coordinate axis).
KeywordsAutomorphism Group Coordinate Axis Connected Graph Edge Form Condition forE
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