TY - GEN

T1 - Mixed connectivity of random graphs

AU - Gu, Ran

AU - Shi, Yongtang

AU - Fan, Neng

N1 - Publisher Copyright:
© Springer International Publishing AG 2017.

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 -