The Colouring problem asks whether the vertices of a graph can be coloured with at most $k$ colours for a given integer $k$ in such a way that no two adjacent vertices receive the same colour. A graph is $(H_1,H_2)$-free if it has no induced subgraph isomorphic to $H_1$ or $H_2$. A connected graph $H_1$ is almost classified if Colouring on $(H_1,H_2)$-free graphs is known to be polynomial-time ...

We consider the allocation problem in which $m \leq (1-\ve) dn $ items are to be allocated to $n$ bins with capacity $d$. The items $x_1,x_2,\ldots,x_m$ arrive sequentially and when item $x_i$ arrives it is given two possible bin locations $p_i=h_1(x_i),q_i=h_2(x_i)$ via hash functions $h_1,h_2$. We consider a random walk procedure for inserting items and show that the expected time insertion t...

The shape space considered in this article consists of surfaces embedded $$\mathbb {R}^3$$ , that are decorated with curves. It is a special case the Fréchet manifolds nonlinear flags, i.e. nested submanifolds fixed type. gauge invariant elastic metric on involves mean curvature and normal deformation, sum difference principal curvatures $$\kappa _1,\kappa _2$$ . proposed metrics curves involve...

We construct an action of the Thompson group $F$ on a compact space built from pairs infinite, binary rooted trees. The arises as $F$-equivariant compactification by translations one its homogeneous spaces, $F/H_2$, corresponding to certain subgroup $H_2$ $F$. representation Hilbert $\ell^2(F/H_2)$ is faithful complex algebra $\mathbb{C}[F]$.

The electron impact ionization cross section of H2 is calculated using a distortedwave approximation. The calculated cross section is compared to the experimental results of Peart and Dolder. The simplicity of the system allows an assessment of the accuracy of the distorted-wave approximation without complications from inaccuracies in the target state, initial continuum state, or final continuu...

A graph $G$ is Ramsey for a $H$ if every colouring of the edges in two colours contains monochromatic copy $H$. Two graphs $H_1$ and $H_2$ are equivalent any only it $H_2$. parameter $s$ distinguishing $s(H_1)\neq s(H_2)$ implies that not equivalent. In this paper we show chromatic number parameter. We also extend to multi-colour case use similar idea find another which distinguishing.