Mixed connectivity properties of random graphs and some special graphs

Ran Gu, Yongtang Shi, Neng Fan

Research output: Contribution to journalArticle

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 languageEnglish (US)
JournalJournal of Combinatorial Optimization
DOIs
StatePublished - Jan 1 2019

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Random Graphs
Connectivity
Graph in graph theory
Hitting Time
Subset
Minimum Degree
Mixed Graphs
Integer
Complete Bipartite Graph
Complete Graph
Connected graph
Generalise

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.

In: Journal of Combinatorial Optimization, 01.01.2019.

Research output: Contribution to journalArticle

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