The matroidal connectivity and conditional matroidal connectivity are novel indicators to measure the real faulty tolerability. In this paper, for the$n$-dimensional alternating group graph$AG_{n}$, the structure properties and (conditional) matroidal connectivity are studied based on the dimensional partition of$E(AG_{n})$. We prove that for$S\subseteq E(AG_{n})$under some limitation on the number of faulty edges in each dimensional edge set, if$|S|\leq (n-1)!-1$, then$AG_{n}-S$is connected. We study the value of matroidal connectivity and conditional matroidal connectivity of$AG_{n}$. Furthermore, simulations have been carried out to compare the matroidal connectivity with other types of conditional connectivity in$AG_{n}$. The simulation result shows that the matroidal connectivity significantly improves these known fault-tolerant capability of alternating group graphs.

The main method to achieve fault-tolerant network systems is by exploiting and effectively utilizing the edge-disjoint and/or inner-vertex-disjoint paths between pairs of source and destination vertices. Completely independent spanning trees (CISTs for short) are powerful tools for reliable broadcasting/unicasting and secure message distribution. Particularly, it has been shown that two CISTs have an application on configuring a protection routing in IP networks, such as mobile ad hoc networks and relatively large (static) network topologies with scalability in [IEEE/ACM Trans. Netw., 27 (2019) 1112-1123]. Many results focus on CISTs in specific networks in the literature, however, few results are given on an infinite class of networks having common properties. In this article, we prove the existence of dual-CISTs in an infinite number of networks satisfying some Hamilton sufficient conditions. A unique algorithm to construct a CIST-partition is proposed, which can be applied to not only many kinds of networks, but our algorithm can also be implemented very easily in parallel or distributed systems satisfying the conditions. In addition, we make a comparative analysis between the proposed conditions and several known results on an infinite number of networks, the advantage of our result is significant. In particular, the bound in our conditions is sharp. The results will provide a powerful framework for the design of fault-tolerant network topologies and routing protocols for future networks.

Given two regular graphs G and H such that the vertex degree of G is equal to the number of vertices in H, the compound graph G(H) is constructed by replacing each vertex of G by a copy of Hand replacing each edge of G by an additional edge connecting random vertices in two corresponding copies of H, respectively, under the constraint that each vertex in G(H) is incident with only one additional edge, exactly. L-HSDCmis a compound graph G(H), where G is a hypercube Qmand H is a complete graph Km, which is defined by focusing on the connected relation between servers in the novel data center network HSDCmproposed in [30]. A set of k spanning trees in a graph G are called completely independent spanning trees (CISTs for short) if the paths joining every pair of vertices x and yin any two trees have neither vertex nor edge in common, except for x and y. In this paper, we give a sufficient condition for the existence of k CISTs in a kind of compound graph. Furthermore, a specific construction algorithm is provided. As corollaries of the main results, the existences of two CISTs form m ≥ 4; three CISTs form m ≥ 8 and four CISTs form m ≥ 10 in L-HSDCm(m) are gotten directly.