In 1999, Jacobson and Lehel conjectured that for \(k \ge 3\), every k-regular Hamiltonian graph has cycles of at least linearly many different lengths. This was further strengthened by Verstraete, who asked whether the regularity can be replaced with weaker condition minimum degree is 3. Despite attention from various researchers, until now best partial result towards both these conjectures a \...

Faudree, R.J., R.J. Gould, M.S. Jacobson and L.M. Lesniak, Neighborhood unions and a generalization of Dirac’s theorem, Discrete Mathematics 105 (1992) 61-71. Dirac proved that if each vertex of a graph G of order n 23 has degree at least n/2, then the graph is Hamiltonian. This result will be generalized by showing that if the union of the neighborhoods of each pair of vertices of a 2connected...

For a graph G a subset D of the vertex set of G is a k-dominating set if every vertex not in D has at least k neighbors in D. The k-domination number γk(G) is the minimum cardinality among the k-dominating sets of G. Note that the 1-domination number γ1(G) is the usual domination number γ(G). Fink and Jacobson showed in 1985 that the inequality γk(G) ≥ γ(G) + k − 2 is valid for every connected ...

We characterize the distributive lattices of Jacobson rings and prove that if a semiring is a distributive lattice of Jacobson rings, then, up to isomorphism, it is equal to the subdirect product of a distributive lattice and a Jacobson ring. Also, we give a general method to construct distributive lattices of Jacobson rings.