By using seismic records of the 2004 magnitude 6.0 Parkfield earthquake, we identified a burst of high-frequency seismic radiation that occurred about 13 kilometers northwest of the hypocenter and 5 seconds after rupture initiation. We imaged this event in three dimensions by using a waveform back-projection method, as well as by timing distinct arrivals visible on many of the seismograms. The ...

Proof: “⇒” Suppose D = (V,A) is a strongly connected graph obtained as an orientation of G = (V,E). Then, since ∀S ⊂ V , |δ D(S)| ≥ 1 and |δ − D(S)| ≥ 1, we have |δG(S)| ≥ 2. Therefore, G is 2-edge-connected. “⇐” Let G be a 2-edge-connected graph. Then G has an ear-decomposition. In other words, G is either a cycle C or G is obtained from a 2-edge-connected graph G by adding an ear P (a path) c...

A k-geodetic digraph G is a digraph in which, for every pair of vertices u and v (not necessarily distinct), there is at most one walk of length ≤ k from u to v. If the diameter of G is k, we say that G is strongly geodetic. Let N(d, k) be the smallest possible order for a k-geodetic digraph of minimum out-degree d, then N(d, k) ≥ 1 + d+ d2 + . . .+ dk = M(d, k), where M(d, k) is the Moore boun...

We consider the dihamiltonian decomposition problem for 3regular graphs. A graphG is dihamiltonian decomposable if in the digraph obtained fromG by replacing each edge of G as two directed edges, the set of edges are partitioned into 3 edge-disjoint directed hamiltonian cycles. We suggest some conditions for dihamiltonian decomposition of 3-regular graphs: for a 3-regular graph G, it is dihamil...

We consider the dihamiltonian decomposition problem for 3-regular graphs. A graph G is dihamiltonian decomposable if in the digraph obtained from G by replacing each edge of G as two directed edges, the set of edges are partitioned into 3 edge-disjoint directed hamiltonian cycles. We suggest some conditions for dihamiltonian decomposition of 3-regular graphs: for a 3-regular graph G, it is diha...

Given graphs G and H, and a colouring of the edges of G with k colours, a monochromatic H-decomposition of G is a partition of the edge set of G such that each part is either a single edge or forms a monochromatic graph isomorphic to H. Let φk(n,H) be the smallest number φ such that any k-edge-coloured graph G of order n, admits a monochromatic H-decomposition with at most φ parts. Here we stud...