Carbon Nanocones with Curvature Effects Close to the Vertex

The conventional rolled-up model for carbon nanocones assumes that the cone is constructed from a rolled-up graphene sheet joined seamlessly, which predicts five distinct vertex angles. This model completely ignores any effects due to the changing curvature, and all bond lengths and bond angles are assumed to be those for the planar graphene sheet. Clearly, curvature effects will become more important closest to the cone vertex, and especially so for the cones with the smaller apex angles. Here, we construct carbon nanocones which, in the assembled cone, are assumed to comprise bond lengths and bond angles that are, as far as possible, equal throughout the structure at the same distance from the conical apex. The predicted bond angles and bond lengths are shown to agree well with those obtained by relaxing the conventional rolled-up model using Lammps software (version: 11 September 2008). The major objective here is not simply to model physically realisable carbon nanocones for which numerical procedures are far superior, but rather, to produce an improved model that takes curvature effects close to the vertex into account, and from which we may determine an analytical formula which represents an improvement on the conventional rolled-up model.

Carbon Nanocones with Curvature Effects Close to Vertex

The conventional rolled-up model for carbon nanocones assumes that the cone is constructed from a rolled-up graphene sheet joined seamlessly, which predicts five distinct vertex angles. This model completely ignores any effects due to the changing curvature and all bond lengths and bond angles are assumed to be those for the planar graphene sheet. Clearly curvature effects will become more important closest to the cone vertex, and especially so for the cones with the smaller apex angles. Here we construct carbon nanocones which in the assembled cone are assumed to comprise bond lengths and bond angles which are, as far as possible, equal throughout the structure at the same distance from the conical apex. Predicted bond angles and bond lengths are shown to agree well with those obtained by relaxing the conventional rolled-up model using the LAMMPS software. The major objective here is not simply to model physically realisable carbon nanocones for which numerical procedures are far superior, but rather to produce an improved model that takes into account curvature effects close to the vertex, and from which we may determine an analytical formula which represents an improvement on that for the conventional rolled-up model.

GUARDING RECTANGULAR PARTITIONS

A rectangular partition is a partition of a rectangle into non-overlapping rectangles, such that no four rectangles meet at a common point. A vertex guard is a guard located at a vertex of the partition (i.e., at a corner of a rectangle); it guards the rectangles that meet at this vertex. An edge guard is a guard that patrols along an edge of the partition, and is thus equivalent to two adjacent vertex guards. We consider the problem of finding a minimum-cardinality guarding set for the rectangles of the partition. For vertex guards, we prove that guarding a given subset of the rectangles is NP-hard. For edge guards, we prove that guarding all rectangles, where guards are restricted to a given subset of the edges, is NP-hard. For both results we show a reduction from vertex cover in non-bipartite 3-connected cubic planar graphs of girth greater than three. For the second NP-hardness result, we obtain a graph-theoretic result which establishes a connection between the set of faces of a plane graph of vertex degree at most three and a vertex cover for this graph. More precisely, we prove that one can assign to each internal face a distinct vertex of the cover, which lies on the face's boundary. We show that the vertices of a rectangular partition can be colored red, green, or black, such that each rectangle has all three colors on its boundary. We conjecture that the above is also true for four colors. Finally, we obtain a worst-case upper bound on the number of edge guards that are sufficient for guarding rectangular partitions with some restrictions on their structure.

FAULT-TOLERANT HAMILTONIAN LACEABILITY AND FAULT-TOLERANT CONDITIONAL HAMILTONIAN FOR BIPARTITE HYPERCUBE-LIKE NETWORKS

A bipartite graph G is hamiltonian laceable if there is a hamiltonian path between any two vertices of G from distinct vertex bipartite sets. A bipartite graph G is k-edge fault-tolerant hamiltonian laceable if G - F is hamiltonian laceable for every F ⊆ E(G) with |F| ≤ k. A graph G is k-edge fault-tolerant conditional hamiltonian if G - F is hamiltonian for every F ⊆ E(G) with |F| ≤ k and δ(G - F) ≥ 2. Let G0 = (V0, E0) and G1 = (V1, E1) be two disjoint graphs with |V0| = |V1|. Let Er = {(v,ɸ(v)) | v ϵ V0,ɸ(v) ϵ V1, and ɸ: V0 → V1 is a bijection}. Let G = G0 ⊕ G1 = (V0 ⋃ V1, E0 ⋃ E1 ⋃ Er). The set of n-dimensional hypercube-like graphHn is defined recursively as (a) H1 = K2, K2 is the complete graph with two vertices, and (b) if G0 and G1 are in Hn, then G = G0 ⊕ G1 is in Hn+1. Let Bn be the set of graphs G where G is bipartite and G ϵ Hn. In this paper, we show that every graph in Bn is (n - 2)-edge fault-tolerant hamiltonian laceable if n ≥ 2 and every graph in Bn is (2n - 5)-edge fault-tolerant conditional hamiltonian if n ≥ 3.