Given two integers n and k, n ≥ k > 1, a k-hypertournament T on n vertices is a pair (V, A), where V is a set of vertices, |V | = n and A is a set of k-tuples of vertices, called arcs, so that for any k-subset S of V , A contains exactly one of the k! k-tuples whose entries belong to S. A 2-hypertournament is merely an (ordinary) tournament. A path is a sequence v1a1v2a2v3...vt−1at−1vt of disti...

The well-known Chvátal–Erdős Theorem states that every graph G of order at least three with α(G) ≤ κ(G) has a hamiltonian cycle, where α(G) and κ(G) are the independence number and the connectivity of G, respectively. Oberly and Sumner [J. Graph Theory 3 (1979), 351–356] have proved that every connected, locally-connected claw-free graph of order at least three has a hamiltonian cycle. We study...

Quantum computing is an important field of research that applies concepts of quantum physics to building more efficient computers. Although only rudimentary quantum computers have been built so far, many researchers believe that quantum computing has great potential and the quantum computers can efficiently perform some tasks which are otherwise not feasible on a classical computer. The Hamilto...

We study the leading term of the holonomy map of a perturbed plane polynomial Hamiltonian foliation. The non-vanishing of this term implies the non-persistence of the corresponding Hamiltonian identity cycle. We prove that this does happen for generic perturbations and cycles, as well for cycles which are commutators in Hamiltonian foliations of degree two. Our approach relies on the Chen’s the...

The classical Lovász conjecture says that every connected Cayley graph is Hamiltonian. We present a short survey of various results in that direction and make some additional observations. In particular, we prove that every finite group G has a generating set of size at most log2 |G|, such that the corresponding Cayley graph contains a Hamiltonian cycle. We also present an explicit construction...

The odd graph Ok has the subsets with k elements of a set {1, . . . , 2k + 1} as its vertices set, and there exists an edge between two vertices if the corresponding pair of k-subsets is disjoint. A conjecture claims that Ok is hamiltonian for k > 2 and another long-standing conjecture implies that all odd graphs have a hamiltonian path. We proved that the prism over Ok is hamiltonian and that ...

Thomassen conjectured [8] that every 4-connected line graph is hamiltonian. An hourglass is a graph isomorphic to K5−E(C4), where C4 is a cycle of length 4 in K5. In [2], it is shown that every 4-connected line graph without an induced subgraph isomorphic to the hourglass is hamiltonian connected. In this note, we prove that every 3-connected, essentially 4-connected hourglass free line graph i...