Abstract
The construction of vertex-disjoint paths (disjoint paths) is an important research topic in various kinds of interconnection networks, which can improve the transmission rate and reliability. The k-ary n-cube is a family of popular networks. In this paper, we determine that there are m2≤m≤n disjoint paths in 3-ary n-cube covering Qn3−F from S to T (many-to-many) with F≤2n−2m and from s to T (one-to-many) with F≤2n−m−1 where s is in a fault-free cycle of length three.
Highlights
Vertex-disjoint paths are a set of paths in a graph that they do not share any vertices
We study m disjoint paths of 3-ary n-cube covering Q3n − F with faulty vertices |F| ≤ 2n − 2m from S to T for |S| |T| m in eorem 1
We consider m disjoint paths of 3-ary n-cube covering Q3n − F with faulty vertices |F| ≤ 2n − m − 1 from s to T where |T| m and s is in a fault-free cycle of length three in eorem 2
Summary
Vertex-disjoint paths (disjoint paths for short) are a set of paths in a graph that they do not share any vertices. A bipartite graph is one-to-one m-disjoint path coverable if there is an m-disjoint path cover between any two vertices in different partite sets. The disjoint path cover problem is extended to a graph with some faulty elements F (vertices and/or edges). We study m disjoint paths of 3-ary n-cube covering Q3n − F with faulty vertices |F| ≤ 2n − 2m from S to T for |S| |T| m in eorem 1. We consider m disjoint paths of 3-ary n-cube covering Q3n − F with faulty vertices |F| ≤ 2n − m − 1 from s to T where |T| m and s is in a fault-free cycle of length three in eorem 2
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