Geodesic
The generalized 3-connectivity and 4-connectivity of crossed cube
Discussiones Mathematicae. Graph Theory, Tome 44 (2024) no. 2, pp. 791-811

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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.
Keywords: crossed cube, internally disjoint trees, generalized $k$-connectivity, fault tolerance 
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Liu, Heqin; Cheng, Dongqin. The generalized 3-connectivity and 4-connectivity of crossed cube. Discussiones Mathematicae. Graph Theory, Tome 44 (2024) no. 2, pp. 791-811. http://geodesic.mathdoc.fr/item/DMGT_2024_44_2_a19/