The Honeycomb toroidal graph is a highly symmetric, vertex-transitive, bipartite graph which has been investigated for certain properties including pan-cyclicity and Hamilton laceability. The main focus of this project was to construct generalised methods for finding Hamilton paths and thus provide a proof of Hamilton laceability for this graph. The resulting proof was successful for a subset o...

In the early Universe matter was crushed to high densities, in a manner similar to that encountered in gravitational collapse to black holes. String theory suggests that the large entropy of black holes can be understood in terms of fractional branes and antibranes. We assume a similar physics for the matter in the early Universe, taking a toroidal compactification and letting branes wrap aroun...

It is shown that toroidal surfaces that extremize a properly weighted surface integral of the squared normal component of a solenoidal three-vector field capture the local invariant dynamics, in that a field line that is anywhere tangential to the surface must be confined to the surface everywhere. In addition to an elementary three-vector calculus derivation, which relies on a curvilinear toro...

We consider the domination number of the queens graph Qn and show that if, for some -xed k, there is a dominating set of Q4k+1 of a certain type with cardinality 2k + 1, then for any n large enough, (Qn)6 [(3k + 5)=(6k + 3)]n + O(1). The same construction shows that for any m¿ 1 and n = 2(6m − 1)(2k + 1) − 1, (Q n)6 [(2k + 3)=(4k + 2)]n + O(1), where Q n is the toroidal n× n queens graph. c © 2...

In this paper we describe the matter-free toroidal spacetime in 't Hooft's polygon approach to 2+1-dimensional gravity (i.e. we consider the case without any particles present). Contrary to earlier results in the literature we nd that it is not possible to describe the torus by just one polygon but we need at least two polygons. We also show that the constraint algebra of the polygons closes.

We show that there is a linear-time algorithm to partition the edges of a planar graph into triangles. We show that the problem is also polynomial for toroidal graphs but NP-complete for k-planar graphs, where k > 8.

The vertex arboricity ρ(G) of a graph G is the minimum number of subsets into which the vertex set V (G) can be partitioned so that each subset induces an acyclic graph. In this paper, it is shown that if G is a toroidal graph without 7-cycles, moreover, G contains no triangular and adjacent 4-cycles, then ρ(G) ≤ 2. Mathematics Subject Classification: Primary: 05C15; Secondary: 05C70

We present a new algorithm to compute periodic (planar) straight-line drawings of toroidal graphs. Our algorithm is the rst to achieve two important aesthetic criteria: the drawing ts in a straight rectangular frame, and the grid area is polynomial, precisely the grid size is O(n×n). This solves one of the main open problems in a recent paper by Duncan et al. [3].

Denote the n× n toroidal queens graph by Qn. We show that γ(Q3k) = k + 2 when k ≡ 0, 3, 4, 6, 8, 9 (mod 12). This completes the proof that γ(Q3k) = 2k − β(Qk) for all positive integers k.

We prove that for m < n, the n × m rectangular toroidal chessboard admits gcd(m,n) nonattacking queens except in the case m = 3, n = 6. The classical n-queens problem is to place n queens on the n × n chessboard such that no pair is attacking each other. Solutions for this problem exist for all for n = 2, 3 [1]. The queens problem on a rectangular board is of little interest; on the n ×m board ...