We study the following Ramsey-type problem. Let S = B ∪ R be a two-colored set of n points in the plane. We show how to construct, in O(n log n) time, a crossing-free spanning tree T (B) for B, and a crossing-free spanning tree T (R) for R, such that both the number of crossings between T (B) and T (R) and the diameters of T (B) and T (R) are kept small. The algorithm is conceptually simple and...

We present a relation between three properties of networks: the fractal properties of the percolation cluster at criticality, the optimal path between vertices in the network under strong disorder (i.e., a broad distribution of edge weights) and the minimum spanning tree. Based on properties of the percolation cluster we show that the distance between vertices under strong disorder and on the m...

Mitigation of atmospheric pollution has been a topic of concern over the past decades. In this study, tree rings have been used to reconstruct past climates as well as to assess the effects of recent climatic and environmental changes on tree growth. Vehicular emission is one of the major sources of pollutants in the atmosphere and this study focused on the Haatso-Atomic road which over the yea...

We prove that the infinite components of the Free Uniform Spanning Forest (FUSF) of a Cayley graph are indistinguishable by any invariant property, given that the forest is different from its wired counterpart. Similar result is obtained for the Free Minimal Spanning Forest (FMSF). We also show that with the above assumptions there can only be 0, 1 or infinitely many components, which solves th...

A classical matrix-tree theorem expresses the determinant of some matrix constructed from a graph (principal minor of the Laplacian) as a sum over all spanning trees of the graph. There are generalizations of this theorem to hypergraphs or simplicial complexes [MV, DKM]. Some version of this theorem provides a formula for the first non-zero coefficient of the Conway polynomial of a (virtual) li...

Hoffmann–Ostenhof’s conjecture states that the edge set of every connected cubic graph can be decomposed into a spanning tree, matching and 2-regular subgraph. In this paper, we show holds for claw-free subcubic graphs 4-chordal graphs.

We study the maximal number of triangulations that a planar set of n points can have, and show that it is at most 30n. This new bound is achieved by a careful optimization of the charging scheme of Sharir and Welzl (2006), which has led to previous best upper bound of 43n for the problem. Moreover, this new bound is useful for bounding the number of other types of planar (i.e., crossing free) s...

We report, to the best of our knowledge, the first demonstration of octave-spanning supercontinuum generation (SCG) on a silicon chip, spanning from the telecommunications c-band near 1.5 μm to the mid-infrared region beyond 3.6 μm. The SCG presented here is characterized by soliton fission and dispersive radiation across two zero group-velocity dispersion wavelengths. In addition, we numerical...

We report the generation of an octave-spanning optical frequency comb in a continuous wave laser pumped microresonator. The generated comb spectrum covers the wavelength range from 990 to 2170 nm without relying on additional external broadening. Continuous tunability of the generated frequency comb over more than an entire free spectral range is demonstrated. Moreover, the linewidth of individ...

We present a protocol for maintaining a spanning tree that is maximal with respect to any given (bounded and monotonic) routing metric. This protocol has two interesting adaptive properties. First, the protocol is stabilizing: starting from any state, the protocol stabilizes to a state where a maximal tree is present. Second, the protocol is loop-free: starting from any state where a spanning t...