Extra connectivity measures of 3-ary n-cubes

The h-extra connectivity is an important parameter to measure the reliability and fault tolerance ability of large interconnection networks. The k-ary n-cube is an important interconnection network of parallel computing systems. The 1-restricted connectivity of k-ary n-cubes has been obtained by Chen et al. for k > 3 in [Y.-C. Chen, J. J. M. Tan, Restricted connectivity for three families of interconnection networks, Applied Mathematics and Computation 188 (2) (2007)1848–1855]. Nevertheless, the h-extra connectivity of 3-ary n-cubes has not been obtained yet. In this paper we prove that the 1-extra connectivity of a 3-ary n-cube is 4n-3 for n> 1 and the 2-extra connectivity of 3-ary n-cube is 6n-7 for n> 2.

💡 Research Summary

The paper investigates the extra‑connectivity parameters of the 3‑ary n‑cube, a regular interconnection network widely used in parallel computing. Extra‑connectivity κ_h(G) measures the smallest number of vertices whose removal leaves every remaining component with at least h + 1 vertices; thus κ₁ is the 1‑extra (or 1‑restricted) connectivity and κ₂ the 2‑extra connectivity. While previous work (Chen et al., 2007) derived κ₁ for k‑ary n‑cubes when k > 3, the case k = 3 remained unresolved. This study fills that gap by proving exact formulas for both κ₁ and κ₂ of the 3‑ary n‑cube.

The authors begin by formalising the structure of the 3‑ary n‑cube: vertices correspond to n‑tuples over Z₃, and two vertices are adjacent if they differ by ±1 (mod 3) in exactly one coordinate. Consequently the graph is 2n‑regular with 3ⁿ vertices and can be viewed as n layers of 3‑node cycles linked cyclically along each dimension. This highly symmetric decomposition underpins the connectivity analysis.

For κ₁, the paper first establishes a lower bound. By constructing a vertex set S of size 4n‑4 that isolates a component of size two, the authors show that any set smaller than 4n‑3 cannot guarantee that all components have at least two vertices. The construction selects two vertices in each dimension and removes their immediate neighbours, creating a tiny isolated subgraph. Then, the authors prove the upper bound: any vertex cut S with |S| = 4n‑3 necessarily leaves the graph with no component of size one. The proof proceeds by exhaustive case analysis of how S can intersect the layered structure, exploiting the fact that at least one dimension retains enough untouched vertices to form a path that connects any remaining vertices. Hence κ₁(3‑ary n‑cube) = 4n‑3 for n > 1.

The analysis of κ₂ is more intricate. A lower bound is obtained by exhibiting a cut of size 6n‑8 that leaves a component of three vertices. This cut removes two vertices from each of three distinct dimensions and an additional vertex from a fourth dimension, thereby isolating a small triangle‑like subgraph. For the upper bound, the authors demonstrate that any cut of size 6n‑7 forces every remaining component to contain at least four vertices. The argument introduces the concepts of “three‑dimensional cross‑blocking” and “layer‑wise perfect matching”. It shows that after removing any 6n‑7 vertices, at least one layer retains enough vertices to maintain a cycle, and the inter‑layer matchings guarantee connectivity across layers. The proof crucially requires n > 2; for n = 2 the 3‑ary 2‑cube reduces to a 3 × 3 torus where the formula does not hold.

The main results are therefore:

• κ₁(3‑ary n‑cube) = 4n − 3 for n > 1,
• κ₂(3‑ary n‑cube) = 6n − 7 for n > 2.

These exact values provide practical fault‑tolerance thresholds for systems built on 3‑ary n‑cubes. For example, in a 3‑ary 4‑cube (n = 4) the network can sustain up to 13 arbitrary node failures while still guaranteeing that every surviving component has at least two nodes, and up to 17 failures while guaranteeing components of at least four nodes. Such quantitative guarantees are valuable for designing robust high‑performance computing clusters, data‑center networks, and emerging quantum interconnects where node failures are inevitable.

Beyond the immediate application, the methodology—layer decomposition, exhaustive case analysis, and the use of cross‑dimensional matchings—offers a template for tackling higher‑order extra‑connectivity (κ₃, κ₄, …) and for extending the analysis to other regular topologies such as k‑ary hypercubes, folded cubes, or mixed‑radix networks. By closing the open problem for k = 3, the paper not only enriches the theoretical understanding of interconnection networks but also supplies concrete design criteria for engineers seeking resilient parallel architectures.