TY - GEN

T1 - Mixed connectivity of random graphs

AU - Gu, Ran

AU - Shi, Yongtang

AU - Fan, Neng

N1 - Funding Information:
Acknowledgement. R. Gu was partially supported by Natural Science Foundation of Jiangsu Province (No. BK20170860), National Natural Science Foundation of China, and Fundamental Research Funds for the Central Universities (No. 2016B14214). Y. Shi was partially supported by the Natural Science Foundation of Tianjin (No. 17JCQNJC00300) and the National Natural Science Foundation of China.

PY - 2017

Y1 - 2017

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)0N, 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)0N, 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

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

A2 - Han, Meng

A2 - Du, Hongwei

A2 - Gao, Xiaofeng

PB - Springer-Verlag

T2 - 11th International Conference on Combinatorial Optimization and Applications, COCOA 2017

Y2 - 16 December 2017 through 18 December 2017

ER -