Complexity and Heegaard genus of an infinite class of compact 3-manifolds

Using the theory of hyperbolic manifolds with totally geodesic boundary, we provide for every integer n greater than 1 a class of such manifolds all having Matveev complexity equal to n and Heegaard genus equal to n+1. All the elements of this class have a single boundary component of genus n, and the numbers of distinct members of the class grows at least exponentially with n.Comment: 15 pages, 7 figure

Chromatic number of Euclidean plane

If the chromatic number of Euclidean plane is larger than four, but it is known that the chromatic number of planar graphs is equal to four, then how does one explain it? In my opinion, they are contradictory to each other. This idea leads to confirm the chromatic number of the plane about its exact value

Palette-colouring: a belief-propagation approach

We consider a variation of the prototype combinatorial-optimisation problem known as graph-colouring. Our optimisation goal is to colour the vertices of a graph with a fixed number of colours, in a way to maximise the number of different colours present in the set of nearest neighbours of each given vertex. This problem, which we pictorially call "palette-colouring", has been recently addressed as a basic example of problem arising in the context of distributed data storage. Even though it has not been proved to be NP complete, random search algorithms find the problem hard to solve. Heuristics based on a naive belief propagation algorithm are observed to work quite well in certain conditions. In this paper, we build upon the mentioned result, working out the correct belief propagation algorithm, which needs to take into account the many-body nature of the constraints present in this problem. This method improves the naive belief propagation approach, at the cost of increased computational effort. We also investigate the emergence of a satisfiable to unsatisfiable "phase transition" as a function of the vertex mean degree, for different ensembles of sparse random graphs in the large size ("thermodynamic") limit.Comment: 22 pages, 7 figure

The algorithmics of solitaire-like games

One-person solitaire-like games are explored with a view to using them in teaching algorithmic problem solving. The key to understanding solutions to such games is the identification of invariant properties of polynomial arithmetic. We demonstrate this via three case studies: solitaire itself, tiling problems and a novel class of one-person games. The known classification of states of the game of (peg) solitaire into 16 equivalence classes is used to introduce the relevance of polynomial arithmetic. Then we give a novel algebraic formulation of the solution to a class of tiling problems. Finally, we introduce an infinite class of challenging one-person games, which we call ``replacement-set games'', inspired by earlier work by Chen and Backhouse on the relation between cyclotomic polynomials and generalisations of the seven-trees-in-one type isomorphism. We present an algorithm to solve arbitrary instances of replacement-set games and we show various ways of constructing infinite (solvable) classes of replacement-set games