The relation between quasiconvexity and k-quasiconvexity, k ≥ 2, is investigated. It is shown that every smooth strictly k-quasiconvex integrand with p-growth at infinity, p > 1, is the restriction to k-th order symmetric tensors of a quasiconvex function with the same growth. When the smoothness condition is dropped, it is possible to prove an approximation result. As a consequence, lower semi...

Let G be a graph with order n, and let k be an integer with 1 ≤ k ≤ n/3. In this article, we show that if σ2(G) ≥ n+ k− 1, then for any stable set S ⊆ V (G) with |S| = k, there exists a 2-factor with precisely k cycles C1, . . . , Ck such that |V (Ci) ∩ S| = 1 for all 1 ≤ i ≤ k and at most one of the cycles Ci has length strictly greater than three. The lower bound on σ2 is best possible.

We prove that any logarithmic binary tree admits a linear-area straight-line strictly-upward planar grid drawing (in short, strictly-upward drawing), that is, a drawing in which (a) each edge is mapped into a single straight-line segment, (b) each node is placed below its parent, (c) no two edges intersect, and (d) each node is mapped into a point with integer coordinates. Informally, a logarit...

We study the online submodular maximization problem with free disposal under a matroid constraint. Elements from some ground set arrive one by one in rounds, and the algorithm maintains a feasible set that is independent in the underlying matroid. In each round when a new element arrives, the algorithmmay accept the new element into its feasible set and possibly remove elements from it, provide...