arXiv subjects
Publications and source records attributed to Narad Rampersad.
A word is cubefree if it contains no non-empty subword of the form xxx. A morphism h : Sigma^* -> Sigma^* is k-uniform if h(a) has length k for all a in Sigma. A morphism is cubefree if it maps cubefree words to cubefree words. We show that for all k >= 0 there exists a k-uniform cubefree binary morphism.
We construct infinite cubefree binary words containing exponentially many distinct squares of length n. We also show that for every positive integer n, there is a cubefree binary square of length 2n.
Given a language L and a nondeterministic finite automaton M, we consider whether we can determine efficiently (in the size of M) if M accepts at least one word in L, or infinitely many words. Given that M accepts at least one word in L, we consider how long a shortest word can be. The languages L that we examine include the palindromes, the non-palindromes, the k-powers, the non-k-powers, the powers, the non-powers (also called primitive words), the words matching a general pattern, the bordered words, and the unbordered words.
Richomme asked the following question: what is the infimum of the real numbers $α$ > 2 such that there exists an infinite word that avoids $α$-powers but contains arbitrarily large squares beginning at every position? We resolve this question in the case of a binary alphabet by showing that the answer is $α$ = 7/3.
For each $α> 2$ there is a binary word with critical exponent $α$.
We give an O(n^3+n^2 t) time algorithm to determine whether an NFA with n states and t transitions accepts a language of polynomial or exponential growth. We also show that given a DFA accepting a language of polynomial growth, we can determine the order of polynomial growth in quadratic time.
We show that the set of binary words containing overlaps is not unambiguously context-free and that the set of ternary words containing overlaps is not context-free. We also show that the set of binary words that are not subwords of the Thue-Morse word is not unambiguously context-free.
Carpi constructed an infinite word over a 4-letter alphabet that avoids squares in all subsequences indexed by arithmetic progressions of odd difference. We show a connection between Carpi's construction and the paperfolding words. We extend Carpi's result by constructing uncountably many words that avoid squares in arithmetic progressions of odd difference. We also construct infinite words avoiding overlaps and infinite words avoiding arbitrarily large squares in arithmetic progressions of odd difference. We use these words to construct labelings of the 2-dimensional integer lattice such that any line through the lattice encounters a squarefree (resp. overlapfree) sequence of labels.
We characterize the squares occurring in infinite overlap-free binary words and construct various alpha power-free binary words containing infinitely many overlaps.
We show that if M is a DFA with n states over an arbitrary alphabet and L = L(M), then the worst-case state complexity of L^2 is n*2^n - 2^{n-1}. If, however, M is a DFA over a unary alphabet, then the worst-case state complexity of L^k is kn-k+1 for all k >= 2.
This paper has been withdrawn by the author due to an error in the proof of Theorem 2.
We examine words w satisfying the following property: if x is a subword of w and |x| is at least k for some fixed k, then the reversal of x is not a subword of w.
In 1982, Seebold showed that the only overlap-free binary words that are the fixed points of non-identity morphisms are the Thue-Morse word and its complement. We strengthen Seebold's result by showing that the same result holds if the term 'overlap-free' is replaced with '7/3-power-free'. Furthermore, the number 7/3 is best possible.
We completely characterize the words that can be avoided in infinite squarefree ternary words.
Alon et al. introduced the concept of non-repetitive colourings of graphs. Here we address some questions regarding non-repetitive colourings of planar graphs. Specifically, we show that the faces of any outerplanar map can be non-repetitively coloured using at most five colours. We also give some lower bounds for the number of colours required to non-repetitively colour the vertices of both outerplanar and planar graphs.
We consider three aspects of avoiding large squares in infinite binary words. First, we construct an infinite binary word avoiding both cubes xxx and squares yy with |y| >= 4; our construction is somewhat simpler than the original construction of Dekking. Second, we construct an infinite binary word avoiding all squares except 00, 11, and 0101; our construction is somewhat simpler than the original construction of Fraenkel and Simpson. In both cases, we also show how to modify our construction to obtain exponentially many words of length n with the given avoidance properties. Finally, we answer an open question of Prodinger and Urbanek from 1979 by demonstrating the existence of two infinite binary words, each avoiding arbitrarily large squares, such that their perfect shuffle has arbitrarily large squares.
Entringer, Jackson, and Schatz conjectured in 1974 that every infinite cubefree binary word contains arbitrarily long squares. In this paper we show this conjecture is false: there exist infinite cubefree binary words avoiding all squares xx with |x| >= 4, and the number 4 is best possible. However, the Entringer-Jackson-Schatz conjecture is true if "cubefree" is replaced with "overlap-free".