3 research outputs found
Tight Bound for the Number of Distinct Palindromes in a Tree
For an undirected tree with edges labelled by single letters, we consider
its substrings, which are labels of the simple paths between pairs of nodes. We
prove that there are different palindromic substrings. This solves
an open problem of Brlek, Lafreni\`ere, and Proven\c{c}al (DLT 2015), who gave
a matching lower-bound construction. Hence, we settle the tight bound of
for the maximum palindromic complexity of trees. For standard
strings, i.e., for paths, the palindromic complexity is . We also propose
-time algorithm for reporting all distinct palindromes in
an undirected tree with edges
Palindromes in starlike trees
In this note, we obtain an upper bound on the maximum number of distinct
non-empty palindromes in starlike trees. This bound implies, in particular,
that there are at most distinct non-empty palindromes in a starlike tree
with three branches each of length . For such starlike trees labelled with a
binary alphabet, we sharpen the upper bound to and conjecture that the
actual maximum is . It is intriguing that this simple conjecture seems
difficult to prove, in contrast to the straightforward proof of the bound.Comment: 5 page
Distinct Squares in Circular Words
A circular word, or a necklace, is an equivalence class under conjugation of
a word. A fundamental question concerning regularities in standard words is
bounding the number of distinct squares in a word of length . The famous
conjecture attributed to Fraenkel and Simpson is that there are at most
such distinct squares, yet the best known upper bound is by Deza et al.
[Discr. Appl. Math. 180, 52-69 (2015)]. We consider a natural generalization of
this question to circular words: how many distinct squares can there be in all
cyclic rotations of a word of length ? We prove an upper bound of .
This is complemented with an infinite family of words implying a lower bound of
.Comment: to appear in SPIRE 201