For a graph Gand an integer k, denote by Vk the set {u E V(G) I d(u) 2 k}. Veldman proved that if G is a 2-connected graph of order n with n 5 3k 2 and IVkl 5 k, then G has a cycle containing all vertices of Vk. It is shown that the upper bound k on IVkl is close to best possible in general. For the special case k = A(G), it is conjectured that the condition lVkl I k can be omitted. Using a var...

Let X be an infinite set of regular cardinality. We determine all clones on X which contain all almost unary functions. It turns out that independently of the size of X, these clones form a countably infinite descending chain. Moreover, all such clones are finitely generated over the unary functions. In particular, we obtain an explicit description of the only maximal clone in this part of the ...

We consider the problem of finding short strings that contain all permutations of order k over an alphabet of size n, with k ≤ n. We show constructively that k(n− 2) + 3 is an upper bound on the length of shortest such strings, for n ≥ k ≥ 10. Consequently, for n ≥ 10, the shortest strings that contain all permutations of order n have length at most n− 2n+ 3. These two new upper bounds improve ...

We consider the problem of finding short strings that contain all permutations of order k over an alphabet of size n, with k ≤ n. We show constructively that k(n− 2) + 3 is an upper bound on the length of shortest such strings, for n ≥ k ≥ 10. Consequently, for n ≥ 10, the shortest strings that contain all permutations of order n have length at most n− 2n+ 3. These two new upper bounds improve ...