### Abstract

Define a minimal detour subgraph of the n-dimensional cube to be a spanning subgraph G of Q_{n} having the property that for vertices x, y of Q_{n}, distances are related by d_{G}(x, y) ≤ d_{Qn}(x, y) + 2. Let f(n) be the minimum number of edges of such a subgraph of Q_{n}. After preliminary work on distances in subgraphs of product graphs, we show that f(n)/|E(Q_{n}| <c/√n. The subgraphs we construct to establish this bound have the property that the longest distances are the same as in Q_{n}, and thus the diameter does not increase. We establish a lower bound for f(n), show that vertices of high degree must be distributed throughout a minimal detour subgraph of Q_{n}, and end with conjectures and questions.

Original language | English |
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Pages (from-to) | 119-128 |

Number of pages | 10 |

Journal | Journal of Graph Theory |

Volume | 23 |

Issue number | 2 |

Publication status | Published - Oct 1996 |

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### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Journal of Graph Theory*,

*23*(2), 119-128.

**Hypercube Subgraphs with Minimal Detours.** / Erdős, P.; Hamburger, Peter; Pippert, Raymond E.; Weakley, William D.

Research output: Contribution to journal › Article

*Journal of Graph Theory*, vol. 23, no. 2, pp. 119-128.

}

TY - JOUR

T1 - Hypercube Subgraphs with Minimal Detours

AU - Erdős, P.

AU - Hamburger, Peter

AU - Pippert, Raymond E.

AU - Weakley, William D.

PY - 1996/10

Y1 - 1996/10

N2 - Define a minimal detour subgraph of the n-dimensional cube to be a spanning subgraph G of Qn having the property that for vertices x, y of Qn, distances are related by dG(x, y) ≤ dQn(x, y) + 2. Let f(n) be the minimum number of edges of such a subgraph of Qn. After preliminary work on distances in subgraphs of product graphs, we show that f(n)/|E(Qn| n, and thus the diameter does not increase. We establish a lower bound for f(n), show that vertices of high degree must be distributed throughout a minimal detour subgraph of Qn, and end with conjectures and questions.

AB - Define a minimal detour subgraph of the n-dimensional cube to be a spanning subgraph G of Qn having the property that for vertices x, y of Qn, distances are related by dG(x, y) ≤ dQn(x, y) + 2. Let f(n) be the minimum number of edges of such a subgraph of Qn. After preliminary work on distances in subgraphs of product graphs, we show that f(n)/|E(Qn| n, and thus the diameter does not increase. We establish a lower bound for f(n), show that vertices of high degree must be distributed throughout a minimal detour subgraph of Qn, and end with conjectures and questions.

UR - http://www.scopus.com/inward/record.url?scp=1542580132&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=1542580132&partnerID=8YFLogxK

M3 - Article

VL - 23

SP - 119

EP - 128

JO - Journal of Graph Theory

JF - Journal of Graph Theory

SN - 0364-9024

IS - 2

ER -