We present a polynomial time algorithm to find the maximum weight of an edge-cut in graphs embeddable on an arbitrary orientable surface, with integral weights bounded in the absolute value by a polynomial of the size of the graph. The algorithm has been implemented for toroidal grids using modular arithmetics and the generalized nested dissection method. The applications in statistical physics...

The game of cops and robbers is a well-known played on graphs. In this paper we consider the straight-ahead orientations 4-regular quadrangulations torus Klein bottle prove that their cop number bounded by constant. We also show every k -regularly oriented toroidal grid at most 13.

Every finite, self-dual, regular (or chiral) 4-polytope of type {3, q, 3} has a trivalent 3-transitive (or 2-transitive) medial layer graph. Here, by dropping self-duality, we obtain a construction for semisymmetric trivalent graphs (which are edgebut not vertex-transitive). In particular, the Gray graph arises as the medial layer graph of a certain universal locally toroidal regular 4-polytope.

This note complements results on cycle spectra of planar graphs by investigating the toroidal case. Let k ≥ 3 be an integer and G a cubic graph polyhedrally embedded torus with circumference at least . For = , it follows from Euler's formula that has some (facial) length in [ 6 ] 4 or 5, we show contains whose lies interval 12 ; for > 5 same holds 2 + is best possible all On other hand, any non...

A k-L(2, 1)-labelling of a graph G is a mapping f : V (G) → {0, 1, 2, . . . , k} such that |f(u)− f(v)| ≥ 2 if uv ∈ E(G) and f(u) 6= f(v) if u, v are distance two apart. The smallest positive integer k such that G admits a k-L(2, 1)-labelling is called the λ-number of G. In this paper we study this quantity for cubic Cayley graphs (other than the prism graphs) on dihedral groups, which are call...

In the present study, the aim is to optimize full and half toroidal continuously variable transmission (CVT) in order to minimize the vehicle fuel consumption (FC) in ECE driving cycle. First, the model of both CVTs’ efficiency is presented. The control strategy of CVTs speed ratio based on minimizing the vehicle FC is introduced, and the algorithm of calculating the vehicle FC is shown. Afterw...

The average effective resistance of a graph can be computed as a function of its eigenvalues. Using this formula and focusing on toroidal d-dimensional graphs, we study the role of the network topology and specifically of the graph dimension in determining the resistance. Considering sequences of graphs of increasing size, we study the asymptotical behavior of the effective resistance, proving ...

The strong chromatic index of a graph is the minimum number of colours needed to colour the edges in such a way that each colour class is an induced matching. In this paper, we present bounds for the strong chromatic index of three different products of graphs in terms of the strong chromatic index of each factor. For the Cartesian product of paths, cycles or complete graphs, we derive sharper ...