The generalized 3-connectivity of the Mycielskian of a graph

Li, Shasha; Zhao, Yan; Li, Fengwei; Gu, Ruijuan

doi:10.1016/j.amc.2018.11.006

Cited by 5 publications

(4 citation statements)

References 17 publications

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“…For any two-bit binary strings x = x 1 x 0 and y = y 1 y 0 , if (x, y) ∈ {(00, 00), (10,10), (11,01), (01, 11)}, then we say they are pair-related, that is x ∼ y [2].…”

Section: Definitions Of Cq N and Related Resultsmentioning

confidence: 99%

“…If z n−1 ∈ {x, y}, suppose z n−1 = y. The proof is similar to Case 1 except that P = yz and 11 n−2 and w n−1 ∈ V CQ 01 n−2 . Two cases will be considered.…”

mentioning

confidence: 84%

“…Since concepts of generalized connectivity were put forward, more and more research results have been published, such as the generalized 3-connectivity of some graphs, including Cartesian product graphs [3], graph products [7], Cayley graphs on symmetric groups generated by trees and cycles [9], star graphs S n and bubble-sort graphs B n [10], the Mycielskian of a graph [11], alternating group graphs and (n, k)-star graph [15], regular graphs with some special properties [17] and so on; the generalized 4-connectivity of some graphs, including hypercubes [12], exchanged hypercubes [16], hierarchical cubic networks [18] and so on. In this paper, we study the generalized 3-connectivity and the generalized 4-connectivity of n-dimensional crossed cube and obtain that κ 3…”

Section: Introductionmentioning

confidence: 99%

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The generalized 3-connectivity and 4-connectivity of crossed cube

Liu¹,

Cheng²

2024

Discuss. Math. Graph Theory

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The generalized connectivity, an extension of connectivity, provides a new reference for measuring the fault tolerance of networks. For any connected graph G, let S ⊆ V (G) and 2 ≤ |S| ≤ V (G); κ G (S) refers to the maximum number of internally disjoint trees in G connecting S. The generalized k-connectivity of G, κ k (G), is defined as the minimum value of κ G (S) over all S ⊆ V (G) with |S| = k. The n-dimensional crossed cube CQ n , as a hypercube-like network, is considered as an attractive alternative to hypercube network because of its many good properties. In this paper, we study the generalized 3-connectivity and the generalized 4-connectivity of CQ n and obtain κ 3 (CQ n ) = κ 4 (CQ n ) = n − 1, where n ≥ 2.

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“…For any two-bit binary strings x = x 1 x 0 and y = y 1 y 0 , if (x, y) ∈ {(00, 00), (10,10), (11,01), (01, 11)}, then we say they are pair-related, that is x ∼ y [2].…”

Section: Definitions Of Cq N and Related Resultsmentioning

confidence: 99%

“…If z n−1 ∈ {x, y}, suppose z n−1 = y. The proof is similar to Case 1 except that P = yz and 11 n−2 and w n−1 ∈ V CQ 01 n−2 . Two cases will be considered.…”

mentioning

confidence: 84%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

The generalized 3-connectivity and 4-connectivity of crossed cube

Liu¹,

Cheng²

2024

Discuss. Math. Graph Theory

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“…In addition, Guo et al [27] derived the Mycielskian graph of a digraph in terms of the vertex(arc) connectivity. Recently, the relationship between the 3-connectivity of G has been established and generalized 3-connectivity of the Mycielskian graph µ(G) has been study by Li et al [28], i.e., κ 3 (µ(G)) ≥ κ 3 (G) + 1. Moreover, they determined the generalized 3-connectivity of the Mycielskian graph of the tree T n , the complete graph K n and the complete bipartite graph K a,b .…”

Section: Introductionmentioning

confidence: 99%

Generalized Connectivity of the Mycielskian Graph under g-Extra Restriction

Zou,

Li,

Zhang

et al. 2023

Mathematics

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The g-extra connectivity is a very important index to evaluate the fault tolerance, reliability of interconnection networks. Let g be a non-negative integer, G be a connected graph with vertex set V and edge set E, a subset S⊆V is called a g-extra cut of G if the graph induced by the set G−S is disconnected and each component of G−S has at least g+1 vertices. The g-extra connectivity of G, denoted as κg(G), is the cardinality of the minimum g-extra cut of G. Mycielski introduced a graph transformation to discover chromatic numbers of triangle-free graphs that can be arbitrarily large. This transformation converts a graph G into a new compound graph called μ(G), also known as the Mycielskian graph of G. In this paper, we study the relationship on g-extra connectivity between the Mycielskian graph μ(G) and the graph G. In addition, we show that κ3(μ(G))=2κ1(G)+1 for κ1(G)≤min{4,⌊n2⌋}, and prove the bounds of κ2g+1(μ(G)) for g≥2.

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