In 2006, Kühn and Osthus showed that if a 3-graph H on n vertices has minimum co-degree at least (1/4 + o(1))n and n is even then H has a loose Hamilton cycle. In this paper, we prove that the minimum co-degree of n/4 suffices. The result is tight.

Let T be a hamiltonian tournament with n vertices and γ a hamiltonian cycle of T . In previous works we introduced and studied the concept of cycle-pancyclism to capture the following question: What is the maximum intersection with γ of a cycle of length k? More precisely, for a cycle Ck of length k in T we denote Iγ(Ck) = |A(γ) ∩ A(Ck)|, the number of arcs that γ and Ck have in common. Let f(k...

A graph G of order n is k-ordered hamiltonian, 2 ≤ k ≤ n, if for every sequence v1, v2, . . . , vk of k distinct vertices of G, there exists a hamiltonian cycle that encounters v1, v2, . . . , vk in this order. In this paper, we generalize two well-known theorems of Chartrand on hamiltonicity of iterated line graphs to k-ordered hamiltonicity. We prove that if Ln(G) is k-ordered hamiltonian and...

In [2], Brousek characterizes all triples of connected graphs, G1, G2, G3, with Gi = K1,3 for some i = 1, 2, or 3, such that all G1G2G3free graphs contain a hamiltonian cycle. In [8], Faudree, Gould, Jacobson and Lesniak consider the problem of finding triples of graphs G1, G2, G3, none of which is a K1,s, s ≥ 3 such that G1G2G3-free graphs of sufficiently large order contain a hamiltonian cycl...

A Hamiltonian cycle in a graph is a cycle that visits each node/vertex exactly once. A graph containing a Hamiltonian cycle is called a Hamiltonian graph. There have been several researches to find the number of Hamiltonian cycles of a Hamilton graph. As the number of vertices and edges grow, it becomes very difficult to keep track of all the different ways through which the vertices are connec...

The basis number b(G) of a graphG is defined to be the least integer k such thatG has a kfold basis for its cycle space. In this note, we determine the basis number of the corona of graphs, in fact we prove that b(v ◦T)= 2 for any tree and any vertex v not inT , b(v ◦H)≤ b(H) + 2, where H is any graph and v is not a vertex of H , also we prove that if G= G1 ◦ G2 is the corona of two graphs G1 a...