Over an alphabet of size 3 we construct an infinite balanced word with critical exponent 2 + √ 2/2. Over an alphabet of size 4 we construct an infinite balanced word with critical exponent (5 + √ 5)/4. Over larger alphabets, we give some candidates for balanced words (found computationally) having small critical exponents. We also explore a method for proving these results using the automated theorem prover Walnut. arXiv:1801.05334v1 [math.CO] 16 Jan 2018 PreliminariesWe let |w| denote the length of a finite word w, and if a is a letter of the alphabet, we let |w| a denote the number of occurrences of a in w.Definition 1. A word w (finite or infinite) over an alphabet A is balanced if for every a ∈ A and every pair u, v of factors of w with |u| = |v| we have | |u| a − |v| a | ≤ 1.Let u be a finite word and write u = u 0 u 1 · · · u n−1 , where the u i are letters. A positive integer p is a period of u if u i = u i+p for all i. Let e = |u|/p and let z be the prefix of u of length p. We say that u has exponent e and write u = z e . The word z is called a fractional root of u. Note that a word may have multiple periods, and consequently, multiple exponents and fractional roots. The word u is primitive if the only integer exponent of u is 1. Let w be a finite or infinite word. The largest r ∈ N (if it exists) such that u r is a factor of w is the (integral) index of u in w.Definition 2. The critical exponent of an infinite word w is E(w) = sup{r ∈ Q : there is a finite, non-empty factor of w with exponent r} = inf{r ∈ Q : there is no finite, non-empty factor of w with exponent r}.The infinite words studied in this paper are constructed by modifying Sturmian words. The structure of such words are determined by a parameter α, which is an irrational real number between 0 and 1, called the slope, and more specifically, by the continued fractionDefinition 3. The characteristic Sturmian word with slope α (see [1, Chapter 9]) is the infinite word c α obtained as the limit of the sequence of standard words s n defined by s 0 = 0, s 1 = 0 d 1 −1 1, s n = s dn n−1 s n−2 , n ≥ 2.
Let G be a simple graph on n vertices. We consider the problem LIS of deciding whether there exists an induced subtree with exactly i ≤ n vertices and leaves in G. We study the associated optimization problem, that consists in computing the maximal number of leaves, denoted by LG(i), realized by an induced subtree with i vertices, for 0 ≤ i ≤ n. We begin by proving that the LIS problem is NP-complete in general and then we compute the values of the map LG for some classical families of graphs and in particular for the d-dimensional hypercubic graphs Q d , for 2 ≤ d ≤ 6. We also describe a nontrivial branch and bound algorithm that computes the function LG for any simple graph G. In the special case where G is a tree of maximum degree ∆, we provide a O(n 3 ∆) time and O(n 2 ) space algorithm to compute the function LG.
We prove that a sequence satisfying a certain symmetry property is $2$-regular in the sense of Allouche and Shallit, i.e., the $\mathbb{Z}$-module generated by its $2$-kernel is finitely generated. We apply this theorem to develop a general approach for studying the $\ell$-abelian complexity of $2$-automatic sequences. In particular, we prove that the period-doubling word and the Thue-Morse word have $2$-abelian complexity sequences that are $2$-regular. Along the way, we also prove that the $2$-block codings of these two words have $1$-abelian complexity sequences that are $2$-regular.
Given a simple graph G with n vertices and a natural number i ≤ n, let L G (i) be the maximum number of leaves that can be realized by an induced subtree T of G with i vertices. We introduce a problem that we call the leaf realization problem, which consists in deciding whether, for a given sequence of n+1 natural numbers ( 0 , 1 , . . . , n ), there exists a simple graph G with n vertices such that i = L G (i) for i = 0, 1, . . . , n. We present basic observations on the structure of these sequences for general graphs and trees. In the particular case where G is a caterpillar graph, we exhibit a bijection between the set of the discrete derivatives of the form (∆L G (i)) 1≤i≤n−3 and the set of prefix normal words.
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