Reliability Evaluation of Augmented Cubes on Degree

The [Formula: see text]-dimensional augmented cube [Formula: see text], proposed by Choudum and Sunitha in 2002, is one of the most famous interconnection networks of the distributed parallel system. Reliability evaluation of underlying topological structures is vital for fault tolerance analysis of this system. As one of the most extensively studied parameters, the [Formula: see text]-conditional edge-connectivity of a connected graph [Formula: see text], [Formula: see text], is defined as the minimum number of the cardinality of the edge-cut of [Formula: see text], if exists, whose removal disconnects this graph and keeps each component of [Formula: see text] having minimum degree at least [Formula: see text]. Let [Formula: see text], [Formula: see text] and [Formula: see text] be three integers, where [Formula: see text], if [Formula: see text] and [Formula: see text], if [Formula: see text]. In this paper, we determine the exact value of the [Formula: see text]-conditional edge-connectivity of [Formula: see text], [Formula: see text] for each positive integer [Formula: see text] and [Formula: see text], and give an affirmative answer to Shinde and Borse’s corresponding conjecture on this topic in [On edge-fault tolerance in augmented cubes, J. Interconnection Netw. 20(4) (2020), DOI:10.1142/S0219265920500139].

The l-good-neighbor edge connectivity of graphs

The l -good-neighbor edge connectivity is an useful parameter to measure the reliability and tolerance of interconnection networks. For a graph H with order p and an integer l (l ≥ 0), an edge subset X ⸦ E(H) is called a l-good-neighbor edge-cut if H − X is disconnected and the minimum degree of every component of H − X is at least £. The order of the minimum l-good-neighbor edge-cut of H is called the l-good-neighbor edge connectivity of H, denoted by λ l (H). In this paper, we show λ(H) ≤ λ l+1(H), obtain the bounds of λl (H) when 0 ≤ l ≤ [p-2/2], character some graphs with the small λl (H) and get some results about the Erdös-Gallai-type problem about λl (H).

The 3-Good Neighbor Edge-Fault-Tolerant Strong Menger Edge Connectivity on the Class of BC Networks

A graph [Formula: see text] is called strongly Menger edge connected (SM-[Formula: see text] for short) if the number of disjoint paths between any two of its vertices equals the minimum degree of these two vertices. In this paper, we focus on the maximally edge-fault-tolerant of the class of BC-networks (contain hypercubes, twisted cubes, Möbius cubes, crossed cubes, etc.) concerning the SM-[Formula: see text] property. Under the restricted condition that each vertex is incident with at least three fault-free edges, we show that even if there are [Formula: see text] faulty edges, all BC-networks still have SM-[Formula: see text] property and the bound [Formula: see text] is sharp.

The (E)FTSM-(edge) Connectivity of Cayley Graphs Generated by Transposition Trees

The Cayley graph generated by a transposition tree [Formula: see text] is a class of Cayley graphs that contains the star graph and the bubble sort graph. A graph [Formula: see text] is called strongly Menger (SM for short) (edge) connected if each pair of vertices [Formula: see text] are connected by [Formula: see text] (edge)-disjoint paths, where [Formula: see text] are the degree of [Formula: see text] and [Formula: see text] respectively. In this paper, the maximally edge-fault-tolerant and the maximally vertex-fault-tolerant of [Formula: see text] with respect to the SM-property are found and thus generalize or improve the results in [19, 20, 22, 26] on this topic.