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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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