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In this paper, we study the connectivity properties of the random subgraph of the n-cube generated by the k-out model and denoted by Qn(k). Let k be an integer, 1 ≤ k ≤ n-1. We let Qn(k) be the graph that is generated by independently including for every ν ∊ V(Qn) a set of k distinct edges chosen uniformly from all the (k n) sets of distinct edges that are incident to ν. We study the connectivity properties of Qn(k) as k varies. We show that without high probability (w.h.p.), Qn(1) does not contain a giant component i.e., a component that spans Ω(2n) vertices. Thereafter, we show that such a component emerges when k=2. In addition, the giant component spans all but o(2n) vertices, and hence it is unique. We then establish the connectivity threshold found at k0=log2 n-2log2log2 n. The threshold is sharp in the sense that Qn([k0]) is disconnected but Qn([k0]+1) is connected w.h.p. Furthermore, we show that w.h.p., Qn(k) is k-connected for every k≥ [k0]+1. [ABSTRACT FROM AUTHOR]

For a simple (unbiased) random walk on a connected graph with n vertices, the cover time (the expected number of steps it takes to visit all vertices) is at most O(n³). We consider locally biased random walks, in which the probability of traversing an edge depends on the degrees of its endpoints. We confirm a conjecture of Abdullah, Cooper, and Draief [2015] that the min-degree local bias rule ensures a cover time of O(n²). For this we formulate and prove the following lemma about spanning trees. Let R(e) denote for edge e the minimum degree among its two endpoints. We say that a weight function W for the edges is feasible if it is nonnegative, dominated by R (for every edge W(e) ≤ R(e)), and the sum over all edges of the ratios W(e)/R(e) equals n - 1. For example, in trees W(e) = R(e), and in regular graphs the sum of edge weights is d(n - 1). We will show in a lemma that for every feasible W, the minimum weight spanning tree has total weight O(n). For regular graphs, a similar lemma was proved by Kahn et al. (1989). [ABSTRACT FROM AUTHOR]

We prove an ear-decomposition theorem for 4-edge-connected graphs and use it to prove that for every 4-edge-connected graph G and every r \in V (G), there is a set of four spanning trees of G with the following property. For every vertex in G, the unique paths back to r in each tree are edge-disjoint. Our proof implies a polynomial-time algorithm for constructing the trees. [ABSTRACT FROM AUTHOR]

We show that every bridgeless cubic graph G with m edges has a cycle cover of length at most 1.6m. Moreover, if G does not contain any intersecting circuits of length 5, then G has a cycle cover of length 212/135 ·m ≈ 1.570m, and if G contains no 5-circuits, then it has a cycle cover of length at most 14/9·m ≈ 1.556m. To prove our results, we show that each 2-edge-connected cubic graph G on n vertices has a 2-factor containing at most n/10+f(G) circuits of length 5, where the value of f(G) depends only on the presence of several subgraphs arising from the Petersen graph. As a corollary we get that each 3-edge-connected cubic graph on n vertices has a 2-factor containing at most n/9 circuits of length 5, and each 4-edge-connected cubic graph on n vertices has a 2-factor containing at most n/10 circuits of length 5. [ABSTRACT FROM AUTHOR]