A sufficient condition for pancyclability of graphs

A new sufficient condition for hamiltonian graphs

The study of Hamiltonian graphs began with Dirac’s classic result in 1952. This was followed by that of Ore in 1960. In 1984 Fan generalized both these results with the following result: If G is a 2-connected graph of order n and max{d(u), d(v)}≥n/2 for each pair of vertices u and v with distance d(u, v)=2, then G is Hamiltonian. In 1991 Faudree–Gould–Jacobson–Lesnick proved that if G is a 2-co...

New sufficient condition for Hamiltonian graphs

Let G be a graph and α(G) be the independence number of G. For a vertex v ∈ V (G), d(v) and N(v) represent the degree of v and the neighborhood of v in G, respectively. In this paper, we prove that if G is a k-connected graph of order n, and if max{d(v) : v ∈ S} ≥ n/2 for every independent set S of G with |S| = k which has two distinct vertices x, y ∈ S satisfying 1 ≤ |N(x) ∩N(y)| ≤ α(G)− 1, th...

A clique-covering sufficient condition for hamiltonicity of graphs

A Sufficient Condition for Intrinsic Knotting of Bipartite Graphs

We present evidence in support of a conjecture that a bipartite graph with at least five vertices in each part and |E(G)| ≥ 4|V (G)| − 17 is intrinsically knotted. We prove the conjecture for graphs that have exactly five or exactly six vertices in one part. We also show that there is a constant Cn such that a bipartite graph with exactly n ≥ 5 vertices in one part and |E(G)| ≥ 4|V (G)| + Cn is...

A Sharp Sufficient Condition for Sparsity Pattern Recovery

Sufficient number of linear and noisy measurements for exact and approximate sparsity pattern/support set recovery in the high dimensional setting is derived. Although this problem as been addressed in the recent literature, there is still considerable gaps between those results and the exact limits of the perfect support set recovery. To reduce this gap, in this paper, the sufficient con...