Mixed connectivity of random graphs

Ran Gu, Yongtang Shi, Neng Fan

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

3 Citations (Scopus)

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
StatePublished - 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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Random Graphs
Connectivity
Hitting Time
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Minimum Degree
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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

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

Combinatorial Optimization and Applications - 11th International Conference, COCOA 2017, Proceedings. Vol. 10627 LNCS Springer Verlag, 2017. p. 133-140 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 10627 LNCS).

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

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, 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
Gu R, Shi Y, Fan N. Mixed connectivity of random graphs. In Combinatorial Optimization and Applications - 11th International Conference, COCOA 2017, Proceedings. Vol. 10627 LNCS. Springer Verlag. 2017. p. 133-140. (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)). https://doi.org/10.1007/978-3-319-71150-8_13
Gu, Ran ; Shi, Yongtang ; Fan, Neng. / Mixed connectivity of random graphs. Combinatorial Optimization and Applications - 11th International Conference, COCOA 2017, Proceedings. Vol. 10627 LNCS Springer Verlag, 2017. pp. 133-140 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)).
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