We pose a new and intriguing question motivated by distributed computing regarding random walks on graphs: How long does it take for several independent random walks, starting from the same vertex, to cover an entire graph? We study the cover time – the expected time required to visit every node in a graph at least once – and we show that for a large collection of interesting graphs, running many random walks in parallel yields a speed-up in the cover time that is linear in the number of parallel walks. We demonstrate that an exponential speed-up is sometimes possible, but that some natural graphs allow only a logarithmic speed-up. A problem related to ours (in which the walks start from some probabilistic distribution on vertices) was previously studied in the context of space efficient algorithms for undirected s–t connectivity and our results yield, in certain cases, an improvement upon some of the earlier bounds.

Let P be a set of n points in ℝ3, and let k ≤ n be an integer. A sphere σ is k-rich with respect to P if |σ ∩ P| ≥ k, and is η-non-degenerate, for a fixed fraction 0 < η < 1, if no circle γ ⊂ σ contains more than η|σ ∩ P| points of P.

We improve the previous bound given in [1] on the number of k-rich η-non-degenerate spheres in 3-space with respect to any set of n points in ℝ3, from O(n4/k5 + n3/k3), which holds for all 0 < η < 1/2, to O*(n4/k11/2 + n2/k2), which holds for all 0 < η < 1 (in both bounds, the constants of proportionality depend on η). The new bound implies the improved upper bound O*(n58/27) ≈ O(n2.1482) on the number of mutually similar triangles spanned by n points in ℝ3; the previous bound was O(n13/6) ≈ O(n2.1667) [1].

We develop lower bounds on the Hadwiger number h(G) of graphs G with high chromatic number. In particular, if G has n vertices and chromatic number k then h(G) ≥ (4k − n)/3.

An r-cut of the complete r-uniform hypergraph Krn is obtained by partitioning its vertex set into r parts and taking all edges that meet every part in exactly one vertex. In other words it is the edge set of a spanning complete r-partite subhypergraph of Krn. An r-cut cover is a collection of r-cuts such that each edge of Krn is in at least one of the cuts. While in the graph case r = 2 any 2-cut cover on average covers each edge at least 2-o(1) times, when r is odd we exhibit an r-cut cover in which each edge is covered exactly once. When r is even no such decomposition can exist, but we can bound the average number of times an edge is cut in an r-cut cover between and . The upper bound construction can be reformulated in terms of a natural polyhedral problem or as a probability problem, and we solve the latter asymptotically.

We prove that the maximum degree Δn of a random series-parallel graph with n vertices satisfies Δn/logn → c in probability, and Δn ~ c logn for a computable constant c > 0. The same kind of result holds for 2-connected series-parallel graphs, for outerplanar graphs, and for 2-connected outerplanar graphs.

We first describe a reduction from the problem of lower-bounding the number of distinct distances determined by a set S of s points in the plane to an incidence problem between points and a certain class of helices (or parabolas) in three dimensions. We offer conjectures involving the new set-up, but are still unable to fully resolve them.

Instead, we adapt the recent new algebraic analysis technique of Guth and Katz [9], as further developed by Elekes, Kaplan and Sharir [6], to obtain sharp bounds on the number of incidences between these helices or parabolas and points in ℝ3. Applying these bounds, we obtain, among several other results, the upper bound O(s3) on the number of rotations (rigid motions) which map (at least) three points of S to three other points of S. In fact, we show that the number of such rotations which map at least k ≥ 3 points of S to k other points of S is close to O(s3/k12/7).

One of our unresolved conjectures is that this number is O(s3/k2), for k ≥ 2. If true, it would imply the lower bound Ω(s/logs) on the number of distinct distances in the plane.

We show that if G is a simple outerplanar graph and H is a graph with the same Tutte polynomial as G, then H is also outerplanar. Examples show that the condition of G being simple cannot be omitted.

We consider a variant of the Cops and Robbers game where the robber can move t edges at a time, and show that in this variant, the cop number of a d-regular graph with girth larger than 2t+2 is Ω(dt). By the known upper bounds on the order of cages, this implies that the cop number of a connected n-vertex graph can be as large as Ω(n2/3) if t ≥ 2, and Ω(n4/5) if t ≥ 4. This improves the Ω() lower bound of Frieze, Krivelevich and Loh (Variations on cops and robbers, J. Graph Theory, to appear) when 2 ≤ t ≤ 6. We also conjecture a general upper bound O(nt/t+1) for the cop number in this variant, generalizing Meyniel's conjecture.

Two players share a connected graph with non-negative weights on the vertices. They alternately take the vertices (one in each turn) and collect their weights. The rule they have to obey is that the remaining part of the graph must be connected after each move. We conjecture that the first player can get at least half of the weight of any tree with an even number of vertices. We provide a strategy for the first player to get at least 1/4 of an even tree. Moreover, we confirm the conjecture for subdivided stars. The parity condition is necessary: Alice gets nothing on a three-vertex path with all the weight at the middle. We suspect a kind of general parity phenomenon, namely, that the first player can gather a substantial portion of the weight of any ‘simple enough’ graph with an even number of vertices.

Let G be a graph with m edges, and let k be a positive integer. We show that V(G) admits a k-partition V1, . . . Vk such that for i ∈ {1, 2, . . . k}, and , where e(Vi) denotes the number of edges with both ends in Vi and . This answers a problem of Bollobás and Scott [2] in the affirmative. Moreover, for i ∈ {1, 2, . . ., k}, which is close to being best possible and settles another problem of Bollobás and Scott [2].