### Abstract

Let X = {x_{1},...,x_{m}} and Y = {y_{1},...,y_{m}} be two disjoints sets of vertices in a graph G. Then (X,Y) is called an antipodal set-pair of size m(m-ASP, for short) if the distance of x_{i} and y_{j} is at most two if and only if i ≠ j. We prove that in a graph of maximum degree k every m-ASP has size m≤ k(k + 1) 2 + 1. This upper bound is nearly best possible since, for every k≥2, there exists a regular graph of degree k, with an m-ASP, m≥ k(k+1) 2 (and m = k(k + 1) 2 + 1 when k ≡ 0 or 1 (mod 4)). If the degrees of the x_{i} and y_{j} are bounded above by p and q, respectively, then an m-ASP can exist only for m≤(^{p + q + 1}_{p}). We conjecture that this bound cán be improved to m≤(^{p} + _{p}^{q}), and verify this conjecture when the graph satisfies some additional assumptions.

Original language | English |
---|---|

Pages (from-to) | 87-93 |

Number of pages | 7 |

Journal | Discrete Mathematics |

Volume | 111 |

Issue number | 1-3 |

DOIs | |

Publication status | Published - Feb 22 1993 |

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### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Theoretical Computer Science

### Cite this

*Discrete Mathematics*,

*111*(1-3), 87-93. https://doi.org/10.1016/0012-365X(93)90144-I

**Bounded degrees and prescribed distances in graphs.** / Caro, Yair; Tuza, Z.

Research output: Contribution to journal › Article

*Discrete Mathematics*, vol. 111, no. 1-3, pp. 87-93. https://doi.org/10.1016/0012-365X(93)90144-I

}

TY - JOUR

T1 - Bounded degrees and prescribed distances in graphs

AU - Caro, Yair

AU - Tuza, Z.

PY - 1993/2/22

Y1 - 1993/2/22

N2 - Let X = {x1,...,xm} and Y = {y1,...,ym} be two disjoints sets of vertices in a graph G. Then (X,Y) is called an antipodal set-pair of size m(m-ASP, for short) if the distance of xi and yj is at most two if and only if i ≠ j. We prove that in a graph of maximum degree k every m-ASP has size m≤ k(k + 1) 2 + 1. This upper bound is nearly best possible since, for every k≥2, there exists a regular graph of degree k, with an m-ASP, m≥ k(k+1) 2 (and m = k(k + 1) 2 + 1 when k ≡ 0 or 1 (mod 4)). If the degrees of the xi and yj are bounded above by p and q, respectively, then an m-ASP can exist only for m≤(p + q + 1p). We conjecture that this bound cán be improved to m≤(p + pq), and verify this conjecture when the graph satisfies some additional assumptions.

AB - Let X = {x1,...,xm} and Y = {y1,...,ym} be two disjoints sets of vertices in a graph G. Then (X,Y) is called an antipodal set-pair of size m(m-ASP, for short) if the distance of xi and yj is at most two if and only if i ≠ j. We prove that in a graph of maximum degree k every m-ASP has size m≤ k(k + 1) 2 + 1. This upper bound is nearly best possible since, for every k≥2, there exists a regular graph of degree k, with an m-ASP, m≥ k(k+1) 2 (and m = k(k + 1) 2 + 1 when k ≡ 0 or 1 (mod 4)). If the degrees of the xi and yj are bounded above by p and q, respectively, then an m-ASP can exist only for m≤(p + q + 1p). We conjecture that this bound cán be improved to m≤(p + pq), and verify this conjecture when the graph satisfies some additional assumptions.

UR - http://www.scopus.com/inward/record.url?scp=43949172432&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=43949172432&partnerID=8YFLogxK

U2 - 10.1016/0012-365X(93)90144-I

DO - 10.1016/0012-365X(93)90144-I

M3 - Article

AN - SCOPUS:43949172432

VL - 111

SP - 87

EP - 93

JO - Discrete Mathematics

JF - Discrete Mathematics

SN - 0012-365X

IS - 1-3

ER -