Path coverings with prescribed ends in faulty hypercubes

We discuss the existence of vertex disjoint path coverings with prescribed ends for the $n$-dimensional hypercube with or without deleted vertices. Depending on the type of the set of deleted vertices and desired properties of the path coverings we establish the minimal integer $m$ such that for every $n \ge m$ such path coverings exist. Using some of these results, for $k \le 4$, we prove Locke's conjecture that a hypercube with $k$ deleted vertices of each parity is Hamiltonian if $n \ge k +2.$ Some of our lemmas substantially generalize known results of I. Havel and T. Dvo\v{r}\'{a}k. At the end of the paper we formulate some conjectures supported by our results.Comment: 26 page

A semi-exact degree condition for Hamilton cycles in digraphs

The paper is concerned with directed versions of Posa's theorem and Chvatal's theorem on Hamilton cycles in graphs. We show that for each a>0, every digraph G of sufficiently large order n whose outdegree and indegree sequences d_1^+ \leq ... \leq d_n^+ and d_1^- \leq >... \leq d_n^- satisfy d_i^+, d_i^- \geq min{i + a n, n/2} is Hamiltonian. In fact, we can weaken these assumptions to (i) d_i^+ \geq min{i + a n, n/2} or d^-_{n - i - a n} \geq n-i; (ii) d_i^- \geq min{i + a n, n/2} or d^+_{n - i - a n} \geq n-i; and still deduce that G is Hamiltonian. This provides an approximate version of a conjecture of Nash-Williams from 1975 and improves a previous result of K\"uhn, Osthus and Treglown