A graph G is pancyclic if G includes cycles of all lengths and G is edge-pancyclic if each edge lies on cycles of all lengths. A bipartite graph is edge-bipancyclic if each edge lies on cycles of every even length from 4 to |V (G)|. Two cycles with the same length m, C1 = ⟨u1, u2, · · · , um, u1⟩ and C2 = ⟨v1, v2, · · · , vm, v1⟩ passing through an edge (x, y) are independent with respect to th...

In this dissertation, we investigate fault-tolerant capabilities of the k-ary n-cubes with respect to the hamiltonian and hamiltonian connected properties. The k-ary n-cube is a bipartite graph if and only if k is an even integer. Let F be a faulty set with nodes and/or links, and let k ≥ 3 be an odd integer. When |F | ≤ 2n − 2, we show that there exists a hamiltonian cycle in a wounded k-ary n...

A graph G on n vertices is Hamiltonian if it contains a spanning cycle, and pancyclic if it contains cycles of all lengths from 3 to n. In 1984, Fan presented a degree condition involving every pair of vertices at distance two for a 2-connected graph to be Hamiltonian. Motivated by Fan’s result, we say that an induced subgraph H of G is f1-heavy if for every pair of vertices u, v ∈ V (H), dH(u,...

We consider only finite undirected graphs without loops or multiple edges. Notation or terms not defined here can be found in [1]. Let G be a graph and let S £; V( G). The subgraph (S) induced by S is the graph with vertex set Sand Whose edge set consists of those edges of G incident with two vertices of S. The distance d(u, v) between vertices u and v in a connected graph G is the minimum numb...

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...

Recently, Chan et al. have proposed a new cycle embedding property called balanced pancyclicity [Discrete Applied Mathematics 155(15) (2007) 1971-1978]. For a graph G(V, E) and any two nodes x and y of V, a cycle R contains x and y can be divided into two paths, Pt1 and Pt2, joining x and y such that len(Pt1)len(Pt2), where len() denote the length of the path . A balanced cycle of an even (r...

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...