R(C4, K4) = 10 (see [2]) R(C4, K5) = 14 (see [3]) R(C5, K4) = 13, R(C5, K5) = 17 (see [5, 6]) R(Cn, K3) = 2n − 1 (n > 3) (see [4, 7]). In [10], we proved that R(Cn, K4) = 3(n − 1) + 1 (n ≥ 4). In this paper, we will prove that R(Cn, K5) = 4(n − 1)+ 1 (n = 6, 7). The following notations will be used in this paper. If G is a graph, the vertex set (resp. edge set) of G is denoted by V (G) (resp. E...

PURPOSE Although the clinical efficacy of cyclosporin A (CSA) in retinoblastoma (RB) has been attributed to multidrug resistance reversal activity, the authors hypothesized that CSA is also directly toxic to RB cells through inhibition of calcineurin (CN)/nuclear factor of activated T-cells (NFAT) signaling. METHODS Antiproliferative effects of CSA, PSC-833 (a CSA analogue that does not inhib...

Let G be an edge-colored graph. A heterochromatic (rainbow, or multicolored) path of G is such a path in which no two edges have the same color. Let CN(v) denote the color neighborhood of a vertex v of G. In a previous paper, we showed that if |CN(u)∪CN(v)| ≥ s (color neighborhood union condition) for every pair of vertices u and v of G, then G has a heterochromatic path of length at least b 2s...

Penta-zirconium copper tribismuth, Zr5CuBi3, crystallizes in the hexa-gonal Hf5CuSn3 structure type. The asymmetric unit contains two Zr sites (site symmetries 3.2 and m2m), one Cu site (site symmetry 3.m) and one Bi site (site symmetry m2m). The environment of the Bi atoms is a tetra-gonal anti-prism with one added atom and a coordination number (CN) of 9. The polyhedron around the Zr1 atom is...

We study the minimal normal completion problem: given A ∈ Cn×n, how do we find an (n+q)×(n+q) normal matrix Aext := ( A A12 A21 A22 ) of smallest possible size? We will show that this smallest number q of rows and columns we need to add, called the normal defect of A, satisfies nd(A) ≥ max{i−(AA∗ −A∗A), i+(AA∗ −A∗A)}, where i±(M) denotes the number of positive and negative eigenvalues of the He...

Let G be a graph whose vertices are labeled 1, . . . , n, and π be a permutation on [n] := {1, 2, . . . , n}. A pebble pi that is initially placed at the vertex i has destination π(i) for each i ∈ [n]. At each step, we choose a matching and swap the two pebbles on each of the edges. Let rt(G, π), the routing number for π, be the minimum number of steps necessary for the pebbles to reach their d...

Let P be a (non necessarily convex) embedded polyhedron in R, with its vertices on an ellipsoid. Suppose that the interior of P can be decomposed into convex polytopes without adding any vertex. Then P is infinitesimally rigid. More generally, let P be a polyhedron bounding a domain which is the union of polytopes C1, · · · , Cn with disjoint interiors, whose vertices are the vertices of P . Su...

Let Wn = K1 ∨ Cn−1 be the wheel graph on n vertices, and let S(n, c, k) be the graph on n vertices obtained by attaching n− 2c− 2k − 1 pendant edges together with k hanging paths of length two at vertex v0, where v0 is the unique common vertex of c triangles. In this paper we show that S(n, c, k) (c > 1, k > 1) and Wn are determined by their signless Laplacian spectra, respectively. Moreover, w...

| Finding vertex-minimal triangulations of closed manifolds is a very difficult problem. Except for spheres and two series of manifolds, vertex-minimal triangulations are known for only few manifolds of dimension more than 2 (see the table given at the end of Section 5). In this article, we present a brief survey on the works done in last 30 years on the following: (i) Finding the minimal numbe...

The vertex-cover problem is studied for random graphs GN,cN having N vertices and cN edges. Exact numerical results are obtained by a branch-and-bound algorithm. It is found that a transition in the coverability at a c-dependent threshold x = xc(c) appears, where xN is the cardinality of the vertex cover. This transition coincides with a sharp peak of the typical numerical effort, which is need...