For d ϵ N, we show that there exists a function f(d) such that every graph G with ∆(G) ≥ f(d) and δ(G) ≥ ⌊d/2⌋ contains every tree on d vertices as a subgraph.

The study of asymptotic minimum degree thresholds that force matchings and tilings in hypergraphs is a lively area of research in combinatorics. A key breakthrough in this area was a result of Hàn, Person, and Schacht [ SIAM J. Disc. Math., 23 (2009), pp. 732--748] who proved that the asymptotic minimum vertex degree threshold for a perfect matching in an $n$-vertex $3$-graph is $\left(\frac{5}{9}+o(1)\right)\binom{n}{2}$. In this paper, we improve on this result, giving a family of degree sequence results, all of which imply the result of Hàn, Person and Schacht and additionally allow one-third of the vertices to have degree $\frac{1}{9}\binom{n}{2}$ below this threshold. Furthermore, we show that this result is, in some sense, tight.

Balogh, Csaba, Jing, and Pluhár [ Electron. J. Combin., 27 (2020)] recently determined the minimum degree threshold that ensures a 2-colored graph $G$ contains a Hamilton cycle of significant color bias (i.e., a Hamilton cycle that contains significantly more than half of its edges in one color). In this short note we extend this result, determining the corresponding threshold for $r$-colorings.

An important result of Komlós [Tiling Turán theorems, Combinatorica, 2000] yields the asymptotically exact minimum degree threshold that ensures a graph $G$ contains an $H$-tiling covering an $x$th proportion of the vertices of $G$ (for any fixed $x \in (0,1)$ and graph $H$). We give a degree sequence strengthening of this result which allows for a large proportion of the vertices in the host graph $G$ to have degree substantially smaller than that required by Komlós's theorem. We also demonstrate that for certain graphs $H$, the degree sequence condition is essentially best possible in more than one sense.

Data storage is an integral part of IT infrastructures, where Quality of Service (QoS) differentiation amongst customers and their applications is essential for many. Achieving this objective in a production environment is nontrivial, because these environments are complex and dynamic. Numerous practical and engineering constraints render the task even more challenging. This paper presents SLED-2, a QoS differentiation solution that meets these challenges in offering effective protection to the performance of important workloads at the expense of less important workloads when needed. SLED-2 uses a customized feedback heuristic that rate-limits selected I/O streams. This approach is unique in that it accounts for a number of important practical considerations, including fine-grained controls, errors in storage systems models, and inexpensive and safe QoS management. SLED-2 has been implemented for the IBM DS8000 series storage servers and shown to be highly effective in a set of hostile and practical scenarios using test facilities for IBM storage products.