Let $G_1,\ldots,G_n$ be graphs on the same vertex set of size $n$, each graph with minimum degree $\delta(G_i)\ge n/2$. A recent conjecture Aharoni asserts that there exists a rainbow Hamiltonian cycle i.e. edge $\{e_1,\ldots,e_n\}$ such $e_i\in E(G_i)$ for $1\leq i \leq n$. This can viewed as version well-known Dirac theorem. In this paper, we prove asymptotically by showing every $\varepsilon...

A graph on n vertices is called pancyclic if it contains a cycle of length ` for all 3 ≤ ` ≤ n. In 1972, Erdős proved that if G is a Hamiltonian graph on n > 4k vertices with independence number k, then G is pancyclic. He then suggested that n = Ω(k) should already be enough to guarantee pancyclicity. Improving on his and some other later results, we prove that there exists a constant c such th...

Abstract Motivated by several conjectures due to Nikoghosyan, in a recent article Li et al., the aim was characterize all possible graphs H such that every 1-tough -free graph is hamiltonian. The almost complete answer given there conclusion proper induced subgraph of $$K_1\cup P_4$$ K 1</mm...

We prove that every 2{connected K 1;3-free and Z 3 ?free graph is hamiltonian except for two graphs. Furthermore, we give a complete characterization of all 2?connected, K 1;3-free graphs, which are not pancyclic, and which are Z 3-free, B-free, W-free, or HP 7 ?free.

Let G be a 2-connected graph of order n. For any u ∈ V (G) and l ∈ {m,m + 1, . . . , n}, if G has a cycle of length l, then G is called [m,n]pancyclic, and if G has a cycle of length l which contains u, then G is called [m,n]-vertex pancyclic. Let δ(G) be a minimum degree ofG and let NG(x) be the neighborhood of a vertex x in G. In [Australas. J. Combin. 12 (1995), 81–91] Liu, Lou and Zhao prov...