### 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 (Publ Math Debrecen 6:290–297, 1959) and Stepanov (Theory Probab Appl 15:55–67, 1970). Furthermore, our argument show that in the random graph process G~=(Gt)0N, 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 (Ann Discrete Math 28:35–38, 1986a; Combinatorica 7:35–38, 1986b) on classic connectivity. Additionally, we obtain the (k, λ) -connectivity and (k, ℓ) -mixed-connectivity of the complete graphs and complete bipartite graphs, and characterize the minimally (k, ℓ) -mixed-connected graphs.

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

Journal | Journal of Combinatorial Optimization |

DOIs | |

State | Published - Jan 1 2019 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Computer Science Applications
- Discrete Mathematics and Combinatorics
- Control and Optimization
- Computational Theory and Mathematics
- Applied Mathematics

### Cite this

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

Research output: Contribution to journal › Article

}

TY - JOUR

T1 - Mixed connectivity properties of random graphs and some special graphs

AU - Gu, Ran

AU - Shi, Yongtang

AU - Fan, Neng

PY - 2019/1/1

Y1 - 2019/1/1

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 (Publ Math Debrecen 6:290–297, 1959) and Stepanov (Theory Probab Appl 15:55–67, 1970). Furthermore, our argument show that in the random graph process G~=(Gt)0N, 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 (Ann Discrete Math 28:35–38, 1986a; Combinatorica 7:35–38, 1986b) on classic connectivity. Additionally, we obtain the (k, λ) -connectivity and (k, ℓ) -mixed-connectivity of the complete graphs and complete bipartite graphs, and characterize the minimally (k, ℓ) -mixed-connected graphs.

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 (Publ Math Debrecen 6:290–297, 1959) and Stepanov (Theory Probab Appl 15:55–67, 1970). Furthermore, our argument show that in the random graph process G~=(Gt)0N, 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 (Ann Discrete Math 28:35–38, 1986a; Combinatorica 7:35–38, 1986b) on classic connectivity. Additionally, we obtain the (k, λ) -connectivity and (k, ℓ) -mixed-connectivity of the complete graphs and complete bipartite graphs, and characterize the minimally (k, ℓ) -mixed-connected graphs.

KW - Connectivity

KW - Edge-connectivity

KW - Hitting time

KW - Mixed connectivity

KW - Random graph

KW - Threshold function

UR - http://www.scopus.com/inward/record.url?scp=85065969722&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85065969722&partnerID=8YFLogxK

U2 - 10.1007/s10878-019-00415-z

DO - 10.1007/s10878-019-00415-z

M3 - Article

JO - Journal of Combinatorial Optimization

JF - Journal of Combinatorial Optimization

SN - 1382-6905

ER -