## An optimal congestion for embedding the hypercube $$H(n)$$ into the line $$P(2^ n)$$. (Congestion optimale du plongement de l’hypercube $$H(n)$$ dans la chaîne $$P(2^ n)$$.)(French)Zbl 0803.68091

Summary: An embedding $$f$$ of a guest graph $$G$$, into a host graph $$H$$, is a one- to-one mapping from each node $$i$$ in $$G$$ to a unique node $$f(i)$$ in $$H$$, and from each edge $$(i,j)$$ in $$G$$ to a path in $$H$$ starting at node $$f(i)$$ and ending at node $$f(j)$$. The congestion of $$f$$ is the maximum number of times any edge of $$H$$ is used by edges of $$G$$. The minimum congestion, over all embeddings, is called the congestion of $$G$$ into $$H$$, and denoted by $$\text{cong}(G,H)$$.
In this paper, we consider the problem of optimally embedding the vertices of hypercube graph $$H(n)$$, in the vertices of a line $$P(2^ n)$$, in order to minimize the congestion; and we show that $\text{cong}(H(n),\;P(2^ n))= 1/3\times (2^{n+1}- 2+ (n\text{ modulo }2)).$ Finally, we conjecture that the value of optimal congestion for embedding hypercube $$H(n)$$ into cycle $$C(2^ n)$$ is $\text{cong}(H(n), C(2^ n))= 1/3\times (5\times 2^{n-2}- 2+ (n\text{ modulo }2)).$ {}.

### MSC:

 68R10 Graph theory (including graph drawing) in computer science 68M07 Mathematical problems of computer architecture 05C10 Planar graphs; geometric and topological aspects of graph theory
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### References:

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