A graph G on n vertices is pancyclic if it contains cycles of length t for all 3 ≤ t ≤ n. In this paper we prove that for any fixed > 0, the random graph G(n, p) with p(n) n−1/2 (i.e., with p(n)/n−1/2 tending to infinity) asymptotically almost surely has the following resilience property. If H is a subgraph of G with maximum degree at most (1/2− )np, then G−H is pancyclic. In fact, we prove a m...

We prove that a k-ary 2-cube Q2 with 3 faulty edges but where every vertex is incident with at least 2 healthy edges is bipancyclic, if k ≥ 3, and k-pancyclic, if k ≥ 5 is odd (these results are optimal).

We study Hamiltonicity and pancyclicity in the graph obtained as union of a deterministic n $$ -vertex H with δ ( ) ≥ α \delta (H)\ge \alpha random d -regular G , for ∈ { 1 2 } d\in \left\{1,2\right\} . When is 2-regular graph, we prove that a.a.s. ∪ H\cup pancyclic all 0 ] \in \left(0,1\right] also extend our result to range sublinear degrees. 1-regular − \left(\sqrt{2}-1,1\right] this best po...

The hypercube family Qn is one of the most well-known interconnection networks in parallel computers. Using Qn, Li et al. introduced dualcube networks DC(n) and proved the vertex symmetry and some fault-tolerant hamiltonian properties of DC(n). Based on the structure of DC(n), Chen and Kao further introduced Dualcube Extensive Networks, denoted by DCEN(G). DCEN(G) is a wider family that contain...

An n-vertex graph is called pancyclic if it contains a cycle of length t for all 3 ≤ t ≤ n. In this paper, we study pancyclicity of random graphs in the context of resilience, and prove that if p n−1/2, then the random graph G(n, p) a.a.s. satisfies the following property: Every Hamiltonian subgraph of G(n, p) with more than (12 +o(1)) ( n 2 ) p edges is pancyclic. This result is best possible ...