Mixed connectivity of random graphs

Ran Gu, Yongtang Shi, Neng Fan

Research output: Chapter in Book/Report/Conference proceedingConference contribution

3 Scopus citations

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 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.

Original languageEnglish (US)
Title of host publicationCombinatorial Optimization and Applications - 11th International Conference, COCOA 2017, Proceedings
PublisherSpringer Verlag
Pages133-140
Number of pages8
Volume10627 LNCS
ISBN (Print)9783319711492
DOIs
Publication statusPublished - Jan 1 2017
Event11th International Conference on Combinatorial Optimization and Applications, COCOA 2017 - Shanghai, China
Duration: Dec 16 2017Dec 18 2017

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume10627 LNCS
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349

Other

Other11th International Conference on Combinatorial Optimization and Applications, COCOA 2017
CountryChina
CityShanghai
Period12/16/1712/18/17

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Keywords

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

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Computer Science(all)

Cite this

Gu, R., Shi, Y., & Fan, N. (2017). Mixed connectivity of random graphs. In 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