Symmetric edge-decompositions of hypercubes |
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Authors: | Mark Ramras |
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Institution: | (1) Department of Mathematics, Northeastern University, 02115 Boston, MA, USA |
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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 j n, a
j
(E) is the number of edges ofE in thej
th
direction (i.e. parallel to thej
th
coordinate axis). |
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Keywords: | |
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