### Abstract

For positive integers k and λ, a graph G is (k,λ) -connected if it satisfies the following two conditions: (1) |V(G)|≥k+1, and (2) for any subset S⊆V(G) and any subset L⊆ E(G) with λ|S|+|L| < kλ, G-(S∪L) is connected. For positive integers k and ℓ, a graph G with |V(G)| ≥ k+ℓ+1 is said to be (k,ℓ)-mixed-connected if for any subset S⊆V(G) and any subset L⊆ E(G) with |S|≤ k,|L|≤ℓ and |S| + |L|< k+ℓ, G-(S∪ L) is connected. In this paper, we investigate the (k, λ) -connectivity and (k,ℓ)-mixed-connectivity of random graphs, and generalize the results of Erdős and Rényi (1959), and Stepanov (1970). Furthermore, our argument can show that in the random graph process G~=(Gt)_{0} ^{N}, N=(n2), the hitting times of minimum degree at least kλ and of G_{t} being (k, λ) -connected coincide with high probability, and also the hitting times of minimum degree at least k+ ℓ and of G_{t} being (k, ℓ)-mixed-connected coincide with high probability. These results are analogous to the work of Bollobás and Thomassen (1986) on classic connectivity.

Original language | English (US) |
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Title of host publication | Combinatorial Optimization and Applications - 11th International Conference, COCOA 2017, Proceedings |

Publisher | Springer Verlag |

Pages | 133-140 |

Number of pages | 8 |

Volume | 10627 LNCS |

ISBN (Print) | 9783319711492 |

DOIs | |

State | Published - Jan 1 2017 |

Event | 11th International Conference on Combinatorial Optimization and Applications, COCOA 2017 - Shanghai, China Duration: Dec 16 2017 → Dec 18 2017 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 10627 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Other

Other | 11th International Conference on Combinatorial Optimization and Applications, COCOA 2017 |
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Country | China |

City | Shanghai |

Period | 12/16/17 → 12/18/17 |

### Fingerprint

### Keywords

- Connectivity
- Edge-connectivity
- Hitting time
- Random graph
- Threshold function

### ASJC Scopus subject areas

- Theoretical Computer Science
- Computer Science(all)

### Cite this

*Combinatorial Optimization and Applications - 11th International Conference, COCOA 2017, Proceedings*(Vol. 10627 LNCS, pp. 133-140). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 10627 LNCS). Springer Verlag. https://doi.org/10.1007/978-3-319-71150-8_13

**Mixed connectivity of random graphs.** / Gu, Ran; Shi, Yongtang; Fan, Neng.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*Combinatorial Optimization and Applications - 11th International Conference, COCOA 2017, Proceedings.*vol. 10627 LNCS, Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 10627 LNCS, Springer Verlag, pp. 133-140, 11th International Conference on Combinatorial Optimization and Applications, COCOA 2017, Shanghai, China, 12/16/17. https://doi.org/10.1007/978-3-319-71150-8_13

}

TY - GEN

T1 - Mixed connectivity of random graphs

AU - Gu, Ran

AU - Shi, Yongtang

AU - Fan, Neng

PY - 2017/1/1

Y1 - 2017/1/1

N2 - For positive integers k and λ, a graph G is (k,λ) -connected if it satisfies the following two conditions: (1) |V(G)|≥k+1, and (2) for any subset S⊆V(G) and any subset L⊆ E(G) with λ|S|+|L| < kλ, G-(S∪L) is connected. For positive integers k and ℓ, a graph G with |V(G)| ≥ k+ℓ+1 is said to be (k,ℓ)-mixed-connected if for any subset S⊆V(G) and any subset L⊆ E(G) with |S|≤ k,|L|≤ℓ and |S| + |L|< k+ℓ, G-(S∪ L) is connected. In this paper, we investigate the (k, λ) -connectivity and (k,ℓ)-mixed-connectivity of random graphs, and generalize the results of Erdős and Rényi (1959), and Stepanov (1970). Furthermore, our argument can show that in the random graph process G~=(Gt)0 N, N=(n2), the hitting times of minimum degree at least kλ and of Gt being (k, λ) -connected coincide with high probability, and also the hitting times of minimum degree at least k+ ℓ and of Gt being (k, ℓ)-mixed-connected coincide with high probability. These results are analogous to the work of Bollobás and Thomassen (1986) on classic connectivity.

AB - For positive integers k and λ, a graph G is (k,λ) -connected if it satisfies the following two conditions: (1) |V(G)|≥k+1, and (2) for any subset S⊆V(G) and any subset L⊆ E(G) with λ|S|+|L| < kλ, G-(S∪L) is connected. For positive integers k and ℓ, a graph G with |V(G)| ≥ k+ℓ+1 is said to be (k,ℓ)-mixed-connected if for any subset S⊆V(G) and any subset L⊆ E(G) with |S|≤ k,|L|≤ℓ and |S| + |L|< k+ℓ, G-(S∪ L) is connected. In this paper, we investigate the (k, λ) -connectivity and (k,ℓ)-mixed-connectivity of random graphs, and generalize the results of Erdős and Rényi (1959), and Stepanov (1970). Furthermore, our argument can show that in the random graph process G~=(Gt)0 N, N=(n2), the hitting times of minimum degree at least kλ and of Gt being (k, λ) -connected coincide with high probability, and also the hitting times of minimum degree at least k+ ℓ and of Gt being (k, ℓ)-mixed-connected coincide with high probability. These results are analogous to the work of Bollobás and Thomassen (1986) on classic connectivity.

KW - Connectivity

KW - Edge-connectivity

KW - Hitting time

KW - Random graph

KW - Threshold function

UR - http://www.scopus.com/inward/record.url?scp=85038217087&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85038217087&partnerID=8YFLogxK

U2 - 10.1007/978-3-319-71150-8_13

DO - 10.1007/978-3-319-71150-8_13

M3 - Conference contribution

AN - SCOPUS:85038217087

SN - 9783319711492

VL - 10627 LNCS

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 133

EP - 140

BT - Combinatorial Optimization and Applications - 11th International Conference, COCOA 2017, Proceedings

PB - Springer Verlag

ER -