Let P be a set of n points, where each point is colored with one the k possible colors. We present efficient algorithms to preprocess such that for given query q, we can quickly identify smallest color spanning object desired type containing q. In this paper, focus on (i) intervals, (ii) axis-parallel square, (iii) rectangle, (iv) equilateral triangle fixed orientation and (v) circle, as our ob...

We show that for every graph G contains two edge-disjoint spanning trees, we can choose trees T1,T2 of such |dT1(v)−dT2(v)|≤5 all v∈V(G). also prove the more general statement positive integer k, there is a constant ck∈O(logk) k T1,…,Tk satisfying |dTi(v)−dTj(v)|≤ck v∈V(G) and i,j∈{1,…,k}. This resolves conjecture Kriesell.

We obtain various sufficient conditions for a graph to have a spanning closed walk meeting each vertex exactly k times or meeting each vertex at most k times. In particular, we generalise the result of Oberly and Sumner that every connected, locally connected K l,3-free graph with at least three vertices is hamil tonian.

Let K/Q be a number field with euclidean ring of integers O. Let Γ be a finite-index torsion-free subgroup of the symplectic group Sp2n(O), and let N be the cohomological dimension of Γ. We exhibit a finite, geometrically-defined spanning set of H (Γ;Z) by generalizing the modular symbol algorithm of Ash and Rudolph for SLn(O).

The number of spanning clusters in four to nine dimensions does not fully follow the expected size dependence for random percolation. Researchers were interested already long ago in percolation theory above the upper critical dimension of six [1, 2], and we followed [3]. At the perco-lation threshold [4], there is a theoretical consensus that the number N of spanning clusters stays finite with ...