### Abstract

Let D ∈ D(n, p) denote a simple random digraph obtained by choosing each of the (n 2) undirected edges independently with probability 2p and then orienting each chosen edge independently in one of the two directions with equal probability 1/2. Let mas(D) denote the maximum size of an induced acyclic subgraph in D. We obtain tight concentration results on the size of mas(D). Precisely, we show that mas(D) ≤ 2/ln(1 - p)^{-1} (ln np + 3e) almost surely, provided p ≥ W/n for some fixed constant W. This combined with known and new lower bounds shows that (for p satisfying p = ω(1/n) and p ≤ 0.5) mas(D) = 2(ln np)/ln(1 - p)^{-1} (1±o(1)). This proves a conjecture stated by Subramanian in 2003 for those p such that p = ω(1/n). Our results are also valid for the random digraph obtained by choosing each of the n(n - 1) directed edges independently with probability p.

Original language | English (US) |
---|---|

Pages (from-to) | 47-54 |

Number of pages | 8 |

Journal | Discrete Mathematics and Theoretical Computer Science |

Volume | 10 |

Issue number | 2 |

State | Published - 2008 |

### Fingerprint

### Keywords

- Induced acyclic subgraphs
- Random digraphs

### ASJC Scopus subject areas

- Computer Science (miscellaneous)
- Computational Mathematics

### Cite this

*Discrete Mathematics and Theoretical Computer Science*,

*10*(2), 47-54.

**On the size of induced acyclic subgraphs in random digraphs.** / Spencer, Joel; Subramanian, C. R.

Research output: Contribution to journal › Article

*Discrete Mathematics and Theoretical Computer Science*, vol. 10, no. 2, pp. 47-54.

}

TY - JOUR

T1 - On the size of induced acyclic subgraphs in random digraphs

AU - Spencer, Joel

AU - Subramanian, C. R.

PY - 2008

Y1 - 2008

N2 - Let D ∈ D(n, p) denote a simple random digraph obtained by choosing each of the (n 2) undirected edges independently with probability 2p and then orienting each chosen edge independently in one of the two directions with equal probability 1/2. Let mas(D) denote the maximum size of an induced acyclic subgraph in D. We obtain tight concentration results on the size of mas(D). Precisely, we show that mas(D) ≤ 2/ln(1 - p)-1 (ln np + 3e) almost surely, provided p ≥ W/n for some fixed constant W. This combined with known and new lower bounds shows that (for p satisfying p = ω(1/n) and p ≤ 0.5) mas(D) = 2(ln np)/ln(1 - p)-1 (1±o(1)). This proves a conjecture stated by Subramanian in 2003 for those p such that p = ω(1/n). Our results are also valid for the random digraph obtained by choosing each of the n(n - 1) directed edges independently with probability p.

AB - Let D ∈ D(n, p) denote a simple random digraph obtained by choosing each of the (n 2) undirected edges independently with probability 2p and then orienting each chosen edge independently in one of the two directions with equal probability 1/2. Let mas(D) denote the maximum size of an induced acyclic subgraph in D. We obtain tight concentration results on the size of mas(D). Precisely, we show that mas(D) ≤ 2/ln(1 - p)-1 (ln np + 3e) almost surely, provided p ≥ W/n for some fixed constant W. This combined with known and new lower bounds shows that (for p satisfying p = ω(1/n) and p ≤ 0.5) mas(D) = 2(ln np)/ln(1 - p)-1 (1±o(1)). This proves a conjecture stated by Subramanian in 2003 for those p such that p = ω(1/n). Our results are also valid for the random digraph obtained by choosing each of the n(n - 1) directed edges independently with probability p.

KW - Induced acyclic subgraphs

KW - Random digraphs

UR - http://www.scopus.com/inward/record.url?scp=46949087253&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=46949087253&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:46949087253

VL - 10

SP - 47

EP - 54

JO - Discrete Mathematics and Theoretical Computer Science

JF - Discrete Mathematics and Theoretical Computer Science

SN - 1365-8050

IS - 2

ER -