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Nonrepetitive Paths and Cycles in Graphs with Application to Sudoku
, 2005
"... Abstract. We provide a simple linear time transformation from a directed or undirected graph with labeled edges to an unlabeled digraph, such that paths in the input graph in which no two consecutive edges have the same label correspond to paths in the transformed graph and vice versa. Using this tr ..."
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complete. We apply our path and cycle finding algorithms in a program for generating and solving Sudoku puzzles, and show experimentally that they lead to effective puzzlesolving rules that may also be of interest to human Sudoku puzzle solvers. 1
Counting, Generating, and Solving Sudoku
, 2008
"... In this work, we give an overview of the research done so far on the topic of Sudoku grid enumeration, solving, and generating Sudoku puzzles. We examine possible extensions and generalizations of previous work on solving and generating Sudoku puzzles focusing mainly on rulebased solvers. A possible ..."
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possible way to influence the difficulty of a generated Sudoku puzzle is described and we introduce new deduction rules for solving a puzzle based on the rules described by David Eppstein in his paper “Nonrepetitive Paths and Cycles in Graphs with Application to Sudoku”. We then generalize these new rules
Notes on nonrepetitive graph colouring
 Electron. J. Combin
, 2008
"... A vertex colouring of a graph is nonrepetitive on paths if there is no path v1, v2,..., v2t such that vi and vt+i receive the same colour for all i = 1, 2,..., t. We determine the maximum density of a graph that admits a kcolouring that is nonrepetitive on paths. We prove that every graph has a sub ..."
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Cited by 8 (5 self)
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A vertex colouring of a graph is nonrepetitive on paths if there is no path v1, v2,..., v2t such that vi and vt+i receive the same colour for all i = 1, 2,..., t. We determine the maximum density of a graph that admits a kcolouring that is nonrepetitive on paths. We prove that every graph has a
NONREPETITIVE COLORINGS OF GRAPHS
"... A sequence a = a1a2...an is said to be nonrepetitive if no two adjacent blocks of a are exactly the same. For instance the sequence 1232321 contains a repetition 2323, while 123132123213 is nonrepetitive. A theorem of Thue asserts that, using only three symbols, one can produce arbitrarily long no ..."
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Cited by 15 (1 self)
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nonrepetitive sequences. In this paper we consider a natural generalization of Thue’s sequences for colorings of graphs. A coloring of the set of edges of a given graph G is nonrepetitive if the sequence of colors on any path in G is nonrepetitive. We call the minimal number of colors needed
Nonrepetitive Colorings of Graphs  A Survey
, 2007
"... A vertex coloring f of a graph G is nonrepetitive if there are no integer r ≥ 1 and a simple path v1,...,v2r in G such that f (vi) = f (vr+i) for all i = 1,...,r. This notion is a graphtheoretic variant of nonrepetitive sequences of Thue. The paper surveys problems and results on this topic. ..."
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Cited by 5 (0 self)
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A vertex coloring f of a graph G is nonrepetitive if there are no integer r ≥ 1 and a simple path v1,...,v2r in G such that f (vi) = f (vr+i) for all i = 1,...,r. This notion is a graphtheoretic variant of nonrepetitive sequences of Thue. The paper surveys problems and results on this topic.
Solving Singledigit Sudoku Subproblems
"... Abstract. We show that singledigit “Nishio ” subproblems in n×n Sudoku puzzles may be solved in time o(2n), faster than previous solutions such as the pattern overlay method. We also show that singledigit deduction in Sudoku is NPhard. 1 ..."
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Abstract. We show that singledigit “Nishio ” subproblems in n×n Sudoku puzzles may be solved in time o(2n), faster than previous solutions such as the pattern overlay method. We also show that singledigit deduction in Sudoku is NPhard. 1
Product geometric crossover for the sudoku puzzle
 In Proceedings of IEEE CEC 2006
, 2006
"... Abstract — Geometric crossover is a representationindependent definition of crossover based on the distance of the search space interpreted as a metric space. It generalizes the traditional crossover for binary strings and other important recombination operators for the most used representations. de ..."
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Cited by 16 (9 self)
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in a simple way. In this paper, we use it to design an evolutionary algorithm to solve the Sudoku puzzle. The different types of constraints make Sudoku an interesting study case for crossover design. We conducted extensive experimental testing and found that, on medium and hard problems, the new
Geometric particle swarm optimization for the sudoku puzzle
 In Proceedings of the Genetic and Evolutionary Computing Conference (GECCO
, 2007
"... Geometric particle swarm optimization (GPSO) is a recently introduced generalization of traditional particle swarm optimization (PSO) that applies to all combinatorial spaces. The aim of this paper is to demonstrate the applicability of GPSO to nontrivial combinatorial spaces. The Sudoku puzzle is ..."
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Cited by 11 (1 self)
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Geometric particle swarm optimization (GPSO) is a recently introduced generalization of traditional particle swarm optimization (PSO) that applies to all combinatorial spaces. The aim of this paper is to demonstrate the applicability of GPSO to nontrivial combinatorial spaces. The Sudoku puzzle
The complexity of nonrepetitive edge coloring of graphs. http://arxiv.org/abs/0709.4497 (accessed February 15th
, 2008
"... A squarefree word is a sequence w of symbols such that there are no strings x, y, and z for which w = xyyz. A nonrepetitive coloring of a graph is an edge coloring in which the sequence of colors along any open path is squarefree. The Thue number π(G) of a graph G is the least n for which the graph ..."
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Cited by 8 (0 self)
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A squarefree word is a sequence w of symbols such that there are no strings x, y, and z for which w = xyyz. A nonrepetitive coloring of a graph is an edge coloring in which the sequence of colors along any open path is squarefree. The Thue number π(G) of a graph G is the least n for which the graph
Deterministically and SudokuDeterministically Recognizable Picture Languages
"... Abstract. The recognizable 2dimensional languages are a robust class with many characterizations, comparable to the regular languages in the 1dimensional case. One characterization is by tiling systems. The corresponding word problem is NPcomplete. Therefore, notions of determinism for tiling sys ..."
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Cited by 2 (0 self)
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systems were suggested. For the notion which was called ”deterministically recognizable ” it was open since 1998 whether it implies recognizability. By showing that acyclicity of grid graphs is recognizable we answer this question positively. In contrast to that, we show that nonrecognizable languages
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