We describe several classes of finite, planar Toeplitz graphs and present results on their chromatic number. We then turn to counting maximal independent sets in these graphs and determine recurrence equations and generating functions for some special cases.

Rectangular lattices are special planar semimodular lattices introduced by G. Grätzer and E. Knapp in 2009. By a patch lattice we mean a rectangular lattice whose weak corners are coatoms. As a sort of gluings, we introduce the concept of a patchwork system. We prove that every glued sum indecomposable planar semimodular lattice is a patchwork of its maximal patch lattice intervals “sewn togeth...

A bstract Feynman integrals in quantum field theory evaluate to special functions and numbers that are usefully described by the notion of transcendental weight. In this paper, we propose a way projecting given dimensionally-regularised integral, for example contributing scattering amplitudes, onto its maximal weight part. The method uses insights into singularity structure space-time loop inte...

The (∆, D) (degree/diameter) problem consists of finding the largest possible number of vertices n among all the graphs with maximum degree ∆ and diameter D. We consider the (∆, D) problem for maximal planar bipartite graphs, that are simple planar graphs in which every face is a quadrangle. We obtain that for the (∆, 2) problem, the number of vertices is n = ∆+2; and for the (∆, 3) problem, n ...

The Hamiltonian cycle problem is one of the most popular NP-complete problems, and remains NP-complete even if we restrict ourselves to a class of (3-connected cubic) planar graphs [5,9]. Therefore, there seems to be no polynomial-time algorithm for the Hamiltonian cycle problem. However, for certain (nontrivial) classes of restricted graphs, there exist polynomial-time algorithms [3,4,6]. In f...

This paper describes tight extended formulations for independent set. The first formulation is for arbitrary independence systems and has size O(n+ μ), where μ denotes the number of inclusion-wise maximal independent sets. Consequently, the extension complexity of the independent set polytope of graphs is O(1.4423). The size O(2n) of the second extended formulation depends on the treewidth tw o...

A hamiltonian walk of a graph is a shortest closed walk that passes through every vertex at least once, and the length is the total number of traversed edges. The hamiltonian walk problem in which one would like to find a hamiltonian walk of a given graph is NP-complete. The problem is a generalized hamiltonian cycle problem and is a special case of the traveling salesman problem. Employing the...

Tahiti Nui is a quiescent tropical volcanic island (French Polynesia) characterized by an intense erosion (up to 0.25 km3·kyr−1 over the last 1 Ma) that resulted in 27 main drainage basins radially distributed around central depression. Those are different rates (from 0.07 24.6 10−2 km3·kyr−1), ages 469 892 ka), micro-climate set-ups and other landform factors. In this study, we assess, terms o...