# Intensive use of computing resources for dominations in grids and other combinatorial problems

Our goal is to prove new results in graph theory and combinatorics thanks to the speed of computers, used with smart algorithms. We tackle four problems. The four-colour theorem states that any map whose countries are connected can be coloured with 4 colours such that neighbouring countries have differnt colours. It was the first result proved using computers, in 1989. We wished to automatise further its proof. We explain the proof and provide a program which replays the proof. It also makes it possible to obtain other results with the same method. We give ideas to automatise the search for discharging rules. We also study the problems of domination in grids. The simplest one is the one of domination. It consists in putting a stone on some cells of a grid such that every cell has a stone, or has a neighbour with a stone. This problem was solved in 2011, using computers to prove a formula giving the minimum number of stones needed. We adapt this method for the first time for variants of the domination. We solve partially two other problems and give for them lower bounds for grids of arbitrary size. We also tackle the counting problem for dominating sets. How many dominating sets are there for a given grid? We study this counting problem for the domination and three variants. For each of these problems, we prove the existence of asymptotic growths rates for which we give bounds. Finally we study polyominoes and the way they can tile rectangles. We tried to solve a problem from 1989: is there a polyomino of odd order? It consists in finding a polyomino which can tile a rectangle with an odd number of copies, but cannot tile any smaller rectangle. We did not manage to solve this problem, but we made a program to enumerate polyominoes and try to find their orders, discarding those which cannot tile rectangles. We also give statistics on the orders of polyominoes of size up to 18.

## Authors

• 3 publications
• ### Asymptotic growth rate of square grids dominating sets: a symbolic dynamics approach

In this text, we prove the existence of an asymptotic growth rate of the...
06/25/2019 ∙ by Silvere Gangloff, et al. ∙ 0

• ### The Connected Domination Number of Grids

Closed form expressions for the domination number of an n × m grid have ...
11/23/2020 ∙ by Adarsh Srinivasan, et al. ∙ 0

• ### Cops and robber on grids and tori

This paper is a contribution to the classical cops and robber problem on...
08/28/2017 ∙ by Fabrizio Luccio, et al. ∙ 0

• ### A method for eternally dominating strong grids

In the eternal domination game, an attacker attacks a vertex at each tur...
03/03/2020 ∙ by Alizée Gagnon, et al. ∙ 0

• ### Counting Answers to Existential Questions

Conjunctive queries select and are expected to return certain tuples fro...
02/13/2019 ∙ by Holger Dell, et al. ∙ 0

• ### Embedding of Hypercube into Cylinder

Task mapping in modern high performance parallel computers can be modele...
11/25/2015 ∙ by Weixing Ji, et al. ∙ 0