2-Disjoint-path-coverable panconnectedness of crossed cubes
The crossed cube is a popular network topology because it possesses many attractive topological properties and its diameter is about half that of the hypercube. Typically, a network topology is modeled as a graph whose vertices and edges represent processors and communication links, respectively. We define a graph \(G\) to be \(2\)-disjoint-path-coverably \(r\)-panconnected for a positive integer \(r\) if for any four distinct vertices \(u,\, v,\, x\), and \(y\) of \(G\), there exist two vertex-disjoint paths \(P_1\) and \(P_2,\) such that (i) \(P_1\) joins \(u\) and \(v\) with length \(l\) for any integer \(l\) satisfying \(r \le l \le |V(G)| - r - 2\), and (ii) \(P_2\) joins \(x\) and \(y\) with length \(|V(G)| - l - 2\), where \(|V(G)|\) is the total number of vertices in \(G\). This property can be considered as an extension of both panconnectedness and connectivity. In this paper, we prove that the \(n\)-dimensional crossed cube is \(2\)-disjoint-path-coverably \(n\)-panconnected.
panconnectedness connectivity crossed cube disjoint path cover interconnection network