### Abstract

Let t(G) be the maximum size of a subset of vertices of a graph G that induces a tree. We investigate the relationship of t(G) to other parameters associated with G: the number of vertices and edges, the radius, the independence number, maximum clique size and connectivity. The central result is a set of upper and lower bounds for the function f(n, ρ{variant}), defined to be the minimum of t(G) over all connected graphs with n vertices and n - 1′ + ρ{variant} edges. The bounds obtained yield an asymptotic characterization of the function correct to leading order in almost all ranges. The results show that f(n, ρ{variant}) is surprisingly small; in particular f(n, cn) = 2 loglogn + O(logloglogn) for any constant c > 0, and f(n, n^{1 + γ}) = 2 log(1 + 1 γ) ± 4 for 0 <γ <1 and n sufficiently large. Bounds on t(G) are obtained in terms of the size of the largest clique. These are used to formulate bounds for a Ramsey-type function, N(k, t), the smallest integer so that every connected graph on N(k, t) vertices has either a clique of size k or an induced tree of size t. Tight bounds for t(G) from the independence number α(G) are also proved. It is shown that every connected graph with radius r has an induced path, and hence an induced tree, on 2r - 1 vertices.

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

Pages (from-to) | 61-79 |

Number of pages | 19 |

Journal | Journal of Combinatorial Theory. Series B |

Volume | 41 |

Issue number | 1 |

DOIs | |

Publication status | Published - 1986 |

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### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Theoretical Computer Science

### Cite this

*Journal of Combinatorial Theory. Series B*,

*41*(1), 61-79. https://doi.org/10.1016/0095-8956(86)90028-6

**Maximum induced trees in graphs.** / Erdős, P.; Saks, Michael; Sós, Vera T.

Research output: Contribution to journal › Article

*Journal of Combinatorial Theory. Series B*, vol. 41, no. 1, pp. 61-79. https://doi.org/10.1016/0095-8956(86)90028-6

}

TY - JOUR

T1 - Maximum induced trees in graphs

AU - Erdős, P.

AU - Saks, Michael

AU - Sós, Vera T.

PY - 1986

Y1 - 1986

N2 - Let t(G) be the maximum size of a subset of vertices of a graph G that induces a tree. We investigate the relationship of t(G) to other parameters associated with G: the number of vertices and edges, the radius, the independence number, maximum clique size and connectivity. The central result is a set of upper and lower bounds for the function f(n, ρ{variant}), defined to be the minimum of t(G) over all connected graphs with n vertices and n - 1′ + ρ{variant} edges. The bounds obtained yield an asymptotic characterization of the function correct to leading order in almost all ranges. The results show that f(n, ρ{variant}) is surprisingly small; in particular f(n, cn) = 2 loglogn + O(logloglogn) for any constant c > 0, and f(n, n1 + γ) = 2 log(1 + 1 γ) ± 4 for 0 <γ <1 and n sufficiently large. Bounds on t(G) are obtained in terms of the size of the largest clique. These are used to formulate bounds for a Ramsey-type function, N(k, t), the smallest integer so that every connected graph on N(k, t) vertices has either a clique of size k or an induced tree of size t. Tight bounds for t(G) from the independence number α(G) are also proved. It is shown that every connected graph with radius r has an induced path, and hence an induced tree, on 2r - 1 vertices.

AB - Let t(G) be the maximum size of a subset of vertices of a graph G that induces a tree. We investigate the relationship of t(G) to other parameters associated with G: the number of vertices and edges, the radius, the independence number, maximum clique size and connectivity. The central result is a set of upper and lower bounds for the function f(n, ρ{variant}), defined to be the minimum of t(G) over all connected graphs with n vertices and n - 1′ + ρ{variant} edges. The bounds obtained yield an asymptotic characterization of the function correct to leading order in almost all ranges. The results show that f(n, ρ{variant}) is surprisingly small; in particular f(n, cn) = 2 loglogn + O(logloglogn) for any constant c > 0, and f(n, n1 + γ) = 2 log(1 + 1 γ) ± 4 for 0 <γ <1 and n sufficiently large. Bounds on t(G) are obtained in terms of the size of the largest clique. These are used to formulate bounds for a Ramsey-type function, N(k, t), the smallest integer so that every connected graph on N(k, t) vertices has either a clique of size k or an induced tree of size t. Tight bounds for t(G) from the independence number α(G) are also proved. It is shown that every connected graph with radius r has an induced path, and hence an induced tree, on 2r - 1 vertices.

UR - http://www.scopus.com/inward/record.url?scp=38249040227&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=38249040227&partnerID=8YFLogxK

U2 - 10.1016/0095-8956(86)90028-6

DO - 10.1016/0095-8956(86)90028-6

M3 - Article

AN - SCOPUS:38249040227

VL - 41

SP - 61

EP - 79

JO - Journal of Combinatorial Theory. Series B

JF - Journal of Combinatorial Theory. Series B

SN - 0095-8956

IS - 1

ER -